In this simulation I will investigate the significance of the regression test. I will simulate 2 different model
Below is the 1st model, which is the Signifianct model \[ Y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \epsilon_i.\]
where \(\epsilon_i\) follows a Normal distribution of mean = 0 and Varience = \(\sigma^2\)
We have provided the value of the parameters as
Below is the 2nd model, which is a Non-Signifianct model \[ Y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \epsilon_i.\]
where \(\epsilon_i\) follows a Normal distribution of mean = 0 and Varience = \(\sigma^2\)
We have provided the value of the parameters as
β3 = 0
n = 25
I will simulate 2500 simulations for both the models for each value of sigma, so around 2500 X 3 = 7500 for each model, so TOTAL 2 X 7500 = 15,000 times for both the model
I need to show there are around 15000 times the model is being trained as part of the simulations
As part of this exercise, the following are the Goals to achieve
In order to achieve these goals, we have various methods in the form of R codes, markdown
Below is the code which will runs the 2500 simulations against each sigma (σ)
birthday = 19770411
set.seed(birthday)
#Variable declaration
n = 25
sigma = c(1,5,10)
no_loop = 2500
#function to generate samples for both significance and non-significance model
mlr_sim = function(n_in,sd=1,signifincance_ind=1){
exer_data = read.csv("study_1.csv")
epsilon = rnorm(n_in,mean = 0,sd = sd)
if (signifincance_ind){
beta_0 = 3
beta_1 = 1
beta_2 = 1
beta_3 = 1
exer_data[,1] = beta_0 + beta_1 * exer_data[,2] + beta_2 * exer_data[,3] + beta_3 * exer_data[,4] + epsilon
}
else {
beta_0 = 3
beta_1 = 0
beta_2 = 0
beta_3 = 0
exer_data[,1] = beta_0 + beta_1 * exer_data[,2] + beta_2 * exer_data[,3] + beta_3 * exer_data[,4] + epsilon
}
exer_data
}
birthday = 19770411
set.seed(birthday)
# Below code is to simulate the Significance Model
model_significance = cbind(sigma_1_F=rep(0,no_loop),sigma_1_P=rep(0,no_loop),sigma_1_R_SQUARED=rep(0,no_loop),
sigma_5_F=rep(0,no_loop),sigma_5_P=rep(0,no_loop),sigma_5_R_SQUARED=rep(0,no_loop),
sigma_10_F=rep(0,no_loop),sigma_10_P=rep(0,no_loop),sigma_10_R_SQUARED=rep(0,no_loop))
extra_counter = 0
#Loop against each sigma to generate the anova for each simulation and populate the model_nsignificance
for(s in 1:length(sigma)){
#print(paste("Sigma:",sigma[s]," extra_counter",extra_counter))
for (i in 1:no_loop){
mlr_data = mlr_sim(n,sigma[s],signifincance_ind = 1)
null_model = lm(y~1,data = mlr_data)
full_model = lm(y~x1+x2+x3,data = mlr_data)
model_significance[i,s+extra_counter] = anova(null_model,full_model)$F[2]
model_significance[i,s+extra_counter+1] = anova(null_model,full_model)$'Pr(>F)'[2]
model_significance[i,s+extra_counter+2] = summary(full_model)$r.squared
}
extra_counter = extra_counter + 2
}
# Below code is to simulate the Non Significance Model
model_nonsignificance = cbind(sigma_1_F=rep(0,no_loop),sigma_1_P=rep(0,no_loop),sigma_1_R_SQUARED=rep(0,no_loop),
sigma_5_F=rep(0,no_loop),sigma_5_P=rep(0,no_loop),sigma_5_R_SQUARED=rep(0,no_loop),
sigma_10_F=rep(0,no_loop),sigma_10_P=rep(0,no_loop),sigma_10_R_SQUARED=rep(0,no_loop))
extra_counter = 0
#Loop against each sigma to generate the anova for each simulation and populate the model_nonsignificance
for(s in 1:length(sigma)){
print(paste("Sigma:",sigma[s]," extra_counter",extra_counter))
for (i in 1:no_loop){
mlr_data = mlr_sim(n,sigma[s],signifincance_ind = 0)
null_model = lm(y~1,data = mlr_data)
full_model = lm(y~x1+x2+x3,data = mlr_data)
model_nonsignificance[i,s+extra_counter] = anova(null_model,full_model)$F[2]
model_nonsignificance[i,s+extra_counter+1] = anova(null_model,full_model)$'Pr(>F)'[2]
model_nonsignificance[i,s+extra_counter+2] = summary(full_model)$r.squared
}
extra_counter = extra_counter + 2
}
## [1] "Sigma: 1 extra_counter 0"
## [1] "Sigma: 5 extra_counter 2"
## [1] "Sigma: 10 extra_counter 4"
As part of the results we will draw out the following
par(mfrow=c(1, 2))
hist(model_significance[,1],
main = "Empirical Distribution of F [Significance]",
cex.main = 0.8,
xlab = "Simulated values of F",
col = "darkolivegreen",
border = "white",
probability = TRUE
)
hist(model_nonsignificance[,1],
main = "Empirical Distribution of F [Non Significance]",
cex.main = 0.8,
xlab = "Simulated values of F",
col = "plum4",
border = "white",
probability = TRUE
)
par(mfrow=c(1, 2))
hist(model_significance[,2],
main = "Empirical Distribution of P [Significance]",
cex.main = 0.8,
xlab = "Simulated values of P",
col = "sienna2",
border = "white",
probability = TRUE
)
hist(model_nonsignificance[,2],
main = "Empirical Distribution of P [Non Significance]",
cex.main = 0.8,
xlab = "Simulated values of P",
col = "slateblue1",
border = "white",
probability = TRUE
)
par(mfrow=c(1, 2))
hist(model_significance[,3],
main = "Empirical Distribution of R Squared [Significance]",
cex.main = 0.7,
xlab = "Simulated values of R Squared",
col = "yellow",
border = "darkolivegreen",
probability = TRUE
)
hist(model_nonsignificance[,3],
main = "Empirical Distribution of R Squared [Non Significance]",
cex.main = 0.7,
xlab = "Simulated values of R Squared",
col = "tan",
border = "white",
probability = TRUE
)
par(mfrow=c(1, 2))
hist(model_significance[,4],
main = "Empirical Distribution of F [Significance Model]",
cex.main = 0.8,
xlab = "Simulated values of F",
col = "darkolivegreen",
border = "white",
probability = TRUE
)
hist(model_nonsignificance[,4],
main = "Empirical Distribution of F [Non Significance]",
cex.main = 0.8,
xlab = "Simulated values of F",
col = "plum4",
border = "white",
probability = TRUE
)
par(mfrow=c(1, 2))
hist(model_significance[,5],
main = "Empirical Distribution of P [Significance]",
cex.main = 0.8,
xlab = "Simulated values of P",
col = "sienna2",
border = "white",
probability = TRUE
)
hist(model_nonsignificance[,5],
main = "Empirical Distribution of P [Non Significance]",
cex.main = 0.8,
xlab = "Simulated values of P",
col = "slateblue1",
border = "white",
probability = TRUE
)
par(mfrow=c(1, 2))
hist(model_significance[,6],
main = "Empirical Distribution of R Squared [Significance]",
cex.main = 0.6,
xlab = "Simulated values of R Squared",
col = "yellow",
border = "white",
probability = TRUE
)
hist(model_nonsignificance[,6],
main = "Empirical Distribution of R Squared [Non Significance]",
cex.main = 0.6,
xlab = "Simulated values of R Squared",
col = "tan",
border = "white",
probability = TRUE
)
par(mfrow=c(1, 2))
hist(model_significance[,7],
main = "Empirical Distribution of F [Significance Model]",
cex.main = 0.6,
xlab = "Simulated values of F",
col = "darkolivegreen",
border = "white",
probability = TRUE
)
hist(model_nonsignificance[,7],
main = "Empirical Distribution of F [Non Significance Model]",
cex.main = 0.6,
xlab = "Simulated values of F",
col = "darkolivegreen",
border = "white",
probability = TRUE
)
par(mfrow=c(1, 2))
hist(model_significance[,8],
main = "Empirical Distribution of P [Significance Model]",
cex.main = 0.6,
xlab = "Simulated values of P",
col = "darkgreen",
border = "white",
probability = TRUE
)
hist(model_nonsignificance[,8],
main = "Empirical Distribution of P [Non Significance Model]",
cex.main = 0.6,
xlab = "Simulated values of P",
col = "darkgreen",
border = "white",
probability = TRUE
)
par(mfrow=c(1, 2))
hist(model_significance[,9],
main = "Empirical Distribution of R Squared [Significance Model]",
cex.main = 0.6,
xlab = "Simulated values of R Squared",
col = "darkcyan",
border = "white",
probability = TRUE
)
hist(model_nonsignificance[,9],
main = "Empirical Distribution of R Squared [Non Significance Model]",
cex.main = 0.6,
xlab = "Simulated values of R Squared",
col = "darkcyan",
border = "white",
probability = TRUE
)
As part of the introduction, we have the following discussion agenda - Do we know the true distribution of the following ? - F statistics - p-value - R square - How do the empirical distributions compare with the true distribution ? - How R2 is related to the sigma
curve(df(x, df1=3, df2=21),
xlab = "Probability",
ylab = "Frequency",
main = "True Distribution of F Test(F Dstribution)",
lwd=3,
col = "darkgreen")
The True distribution of the F Tests shows as a F distribution, which is right skewed distribution
hist(pnorm(rnorm(25,mean = 3, sd = 1),mean=3,sd=1),
xlab = "Probability",
ylab = "Desnity",
main = "True Distribution of P val (Uniform Distribution)",
breaks = 10,
col = "orange",
border = "white",
probability = TRUE)
The True distribution of the P seems to have an uniform distribution
curve(dbeta(x, (4-1)/2, (25-4)/2),
xlab = "Probability",
ylab = "Frequency",
main = "True Distribution of R Squared (Beta distribution)",
lwd = 3,
col = "palevioletred4")
The True distribution of R Squared is a beta distribution (right skewed distribution), where the shape 1 and shape 2 parameter is of value (k-1)/2 and (n-k)/2, where k = number of esimator in the True model(which is 4,i.e \(\beta_{0} , \beta_{1} , \beta_{2} , \beta_{3}\)) and n = number of elements in the data (which is 25)
In this simulation project we will investigate the procedure for choosing the best model via test/train RMSE by simulating the following multiple regression model (MLR)
\[ Y_i = \beta_0 + \beta_1 x_{i1} + \beta_2 x_{i2} + \beta_3 x_{i3} + \beta_4 x_{i4} + \beta_5 x_{i5} + \beta_6 x_{i6} + \epsilon_i.\]
where \(\epsilon_i\) follows a Normal distribution of mean = 0 and Varience = \(\sigma^2\)
We have provided the value of the parameters as
β6 = 0.3
n = 500
We need to evaluate 9 models (below), where for each model, we will calculate the RMSE for train data set (250 random observation) and test data set (250 random observation). We will simulate this for 1000 iteration for each value of σ, so there would be of 3000 simulation for each model, which equals to 3 X 3 X 3000 = 27000 for around 9 models
The Train/Test RMSE for each model is as below
RMSE(model, data) = \(\sqrt{\dfrac{1}{n}\sum\limits_{t=1}^{n}(y_{i} - \hat{y}_i)^2}\)
As per the above model, the correct model is the Model -6, we need to see whether by this methods/approach, we will able to get the correct model
As part of this exercise, the following are the Goals to achieve
Discuss on the following points in the Disucssion section
In order to achieve these goals, we have various methods in the form of R codes, markdown
Below is the code which will runs the 1000 simulations against each sigma (σ)
birthday = 19770411
set.seed(birthday)
#Declaration
beta_0 = 0
beta_1 = 5
beta_2 = -4
beta_3 = 1.6
beta_4 = -1.1
beta_5 = 0.7
beta_6 = 0.3
n = 500
no_loop = 1000
sigma = c(1,2,4)
sig_data = read.csv("study_2.csv")
#Function to generate the sample data
mlr_sim = function(data,sd=1){
epsilon = rnorm(n,mean = 0,sd = sd)
data[,1] = beta_0 + beta_1 * data[,2] + beta_2 * data[,3] + beta_3 * data[,4] + beta_4 * data[,5] + beta_5 * data[,6] + beta_6 * data[,7] + epsilon
data
}
#Defining the Train Matrix to stored the TRAIN RMSE
train_rmse_model1_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model2_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model3_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model4_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model5_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model6_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model7_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model8_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_model9_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
#Defining the Test Matrix to stored the TESE RMSE
test_rmse_model1_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model2_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model3_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model4_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model5_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model6_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model7_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model8_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
test_rmse_model9_mat = cbind(sigma_1=rep(0,no_loop),sigma_2=rep(0,no_loop),sigma_4=rep(0,no_loop))
train_rmse_mat = cbind(sigma_1=rep(0,9),sigma_2=rep(0,9),sigma_4=rep(0,9))
test_rmse_mat = cbind(sigma_1=rep(0,9),sigma_2=rep(0,9),sigma_4=rep(0,9))
#model counter variable to increase the index of the column in the matrix
model1_counter_all_sigma = 0
model2_counter_all_sigma = 0
model3_counter_all_sigma = 0
model4_counter_all_sigma = 0
model5_counter_all_sigma = 0
model6_counter_all_sigma = 0
model7_counter_all_sigma = 0
model8_counter_all_sigma = 0
model9_counter_all_sigma = 0
#Looping through the each sigma to simulate for each 9 models
for(s in 1:length(sigma)){
for(i in 1:no_loop){
#epsilon = rnorm(n,mean = 0, sd = sigma[s])
mlr_data=mlr_sim(sig_data,sd=sigma[s])
#Split the Data into Train Data (Random)
trn_idx = sample(1:nrow(mlr_data), 250)
n_test = nrow(mlr_data) - length(trn_idx)
#Model Creation
model1 = lm(y~x1,data=mlr_data[trn_idx,])
model2 = lm(y~x1+x2,data=mlr_data[trn_idx,])
model3 = lm(y~x1+x2+x3,data=mlr_data[trn_idx,])
model4 = lm(y~x1+x2+x3+x4,data=mlr_data[trn_idx,])
model5 = lm(y~x1+x2+x3+x4+x5,data=mlr_data[trn_idx,])
model6 = lm(y~x1+x2+x3+x4+x5+x6,data=mlr_data[trn_idx,])
model7 = lm(y~x1+x2+x3+x4+x5+x6+x7,data=mlr_data[trn_idx,])
model8 = lm(y~x1+x2+x3+x4+x5+x6+x7+x8,data=mlr_data[trn_idx,])
model9 = lm(y~x1+x2+x3+x4+x5+x6+x7+x8+x9,data=mlr_data[trn_idx,])
#Predict the Test Data for th 9 model
newdata_model1 = subset(mlr_data[-trn_idx,],select=c("x1"))
newdata_model2 = subset(mlr_data[-trn_idx,],select=c("x1","x2"))
newdata_model3 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3"))
newdata_model4 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4"))
newdata_model5 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5"))
newdata_model6 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5","x6"))
newdata_model7 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5","x6","x7"))
newdata_model8 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5","x6","x7","x8"))
newdata_model9 = subset(mlr_data[-trn_idx,],select=c("x1","x2","x3","x4","x5","x6","x7","x8","x9"))
#Train RMSE
train_rmse_model1 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model1))^2) / length(trn_idx))
train_rmse_model2 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model2))^2) / length(trn_idx))
train_rmse_model3 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model3))^2) / length(trn_idx))
train_rmse_model4 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model4))^2) / length(trn_idx))
train_rmse_model5 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model5))^2) / length(trn_idx))
train_rmse_model6 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model6))^2) / length(trn_idx))
train_rmse_model7 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model7))^2) / length(trn_idx))
train_rmse_model8 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model8))^2) / length(trn_idx))
train_rmse_model9 = sqrt(sum((mlr_data[trn_idx,]$y - predict(model9))^2) / length(trn_idx))
train_rmse_model1_mat[i,s] = train_rmse_model1
train_rmse_model2_mat[i,s] = train_rmse_model2
train_rmse_model3_mat[i,s] = train_rmse_model3
train_rmse_model4_mat[i,s] = train_rmse_model4
train_rmse_model5_mat[i,s] = train_rmse_model5
train_rmse_model6_mat[i,s] = train_rmse_model6
train_rmse_model7_mat[i,s] = train_rmse_model7
train_rmse_model8_mat[i,s] = train_rmse_model8
train_rmse_model9_mat[i,s] = train_rmse_model9
#Test RMSE
test_rmse_model1 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model1,newdata=newdata_model1))^2) / n_test)
test_rmse_model2 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model2,newdata=newdata_model2))^2) / n_test)
test_rmse_model3 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model3,newdata=newdata_model3))^2) / n_test)
test_rmse_model4 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model4,newdata=newdata_model4))^2) / n_test)
test_rmse_model5 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model5,newdata=newdata_model5))^2) / n_test)
test_rmse_model6 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model6,newdata=newdata_model6))^2) / n_test)
test_rmse_model7 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model7,newdata=newdata_model7))^2) / n_test)
test_rmse_model8 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model8,newdata=newdata_model8))^2) / n_test)
test_rmse_model9 = sqrt(sum((mlr_data[-trn_idx,]$y - predict(model9,newdata=newdata_model9))^2) / n_test)
test_rmse_model1_mat[i,s] = test_rmse_model1
test_rmse_model2_mat[i,s] = test_rmse_model2
test_rmse_model3_mat[i,s] = test_rmse_model3
test_rmse_model4_mat[i,s] = test_rmse_model4
test_rmse_model5_mat[i,s] = test_rmse_model5
test_rmse_model6_mat[i,s] = test_rmse_model6
test_rmse_model7_mat[i,s] = test_rmse_model7
test_rmse_model8_mat[i,s] = test_rmse_model8
test_rmse_model9_mat[i,s] = test_rmse_model9
#Increase the Model Counter
model1_counter_all_sigma = model1_counter_all_sigma + 1
model2_counter_all_sigma = model2_counter_all_sigma + 1
model3_counter_all_sigma = model3_counter_all_sigma + 1
model4_counter_all_sigma = model4_counter_all_sigma + 1
model5_counter_all_sigma = model5_counter_all_sigma + 1
model6_counter_all_sigma = model6_counter_all_sigma + 1
model7_counter_all_sigma = model7_counter_all_sigma + 1
model8_counter_all_sigma = model8_counter_all_sigma + 1
model9_counter_all_sigma = model9_counter_all_sigma + 1
}
}
#Print the message of Number of times the each model (1/2/3/4/5/6/7/8/9) is being trained for each value of sigma
print (paste("Number of times the model : 1 trained for alpha:",sigma[s]," is: ",model1_counter_all_sigma))
## [1] "Number of times the model : 1 trained for alpha: 4 is: 3000"
print (paste("Number of times the model : 2 trained for alpha:",sigma[s]," is: ",model2_counter_all_sigma))
## [1] "Number of times the model : 2 trained for alpha: 4 is: 3000"
print (paste("Number of times the model : 3 trained for alpha:",sigma[s]," is: ",model3_counter_all_sigma))
## [1] "Number of times the model : 3 trained for alpha: 4 is: 3000"
print (paste("Number of times the model : 4 trained for alpha:",sigma[s]," is: ",model4_counter_all_sigma))
## [1] "Number of times the model : 4 trained for alpha: 4 is: 3000"
print (paste("Number of times the model : 5 trained for alpha:",sigma[s]," is: ",model5_counter_all_sigma))
## [1] "Number of times the model : 5 trained for alpha: 4 is: 3000"
print (paste("Number of times the model : 6 trained for alpha:",sigma[s]," is: ",model6_counter_all_sigma))
## [1] "Number of times the model : 6 trained for alpha: 4 is: 3000"
print (paste("Number of times the model : 7 trained for alpha:",sigma[s]," is: ",model7_counter_all_sigma))
## [1] "Number of times the model : 7 trained for alpha: 4 is: 3000"
print (paste("Number of times the model : 8 trained for alpha:",sigma[s]," is: ",model8_counter_all_sigma))
## [1] "Number of times the model : 8 trained for alpha: 4 is: 3000"
print (paste("Number of times the model : 9 trained for alpha:",sigma[s]," is: ",model9_counter_all_sigma))
## [1] "Number of times the model : 9 trained for alpha: 4 is: 3000"
#Average out the Train RMSE / Test RMSE for each value of sigma
for(s in 1:length(sigma)){
train_rmse_mat[1,s] = mean(train_rmse_model1_mat[,s])
train_rmse_mat[2,s] = mean(train_rmse_model2_mat[,s])
train_rmse_mat[3,s] = mean(train_rmse_model3_mat[,s])
train_rmse_mat[4,s] = mean(train_rmse_model4_mat[,s])
train_rmse_mat[5,s] = mean(train_rmse_model5_mat[,s])
train_rmse_mat[6,s] = mean(train_rmse_model6_mat[,s])
train_rmse_mat[7,s] = mean(train_rmse_model7_mat[,s])
train_rmse_mat[8,s] = mean(train_rmse_model8_mat[,s])
train_rmse_mat[9,s] = mean(train_rmse_model9_mat[,s])
test_rmse_mat[1,s] = mean(test_rmse_model1_mat[,s])
test_rmse_mat[2,s] = mean(test_rmse_model2_mat[,s])
test_rmse_mat[3,s] = mean(test_rmse_model3_mat[,s])
test_rmse_mat[4,s] = mean(test_rmse_model4_mat[,s])
test_rmse_mat[5,s] = mean(test_rmse_model5_mat[,s])
test_rmse_mat[6,s] = mean(test_rmse_model6_mat[,s])
test_rmse_mat[7,s] = mean(test_rmse_model7_mat[,s])
test_rmse_mat[8,s] = mean(test_rmse_model8_mat[,s])
test_rmse_mat[9,s] = mean(test_rmse_model9_mat[,s])
}
As we have discussed in the Introduction on the aspect of Goal, we have to producded plots to support these Goals
plot(seq(1,9), train_rmse_mat[,1], type="o", col="blue", pch="o", lty=1,main="Train, Test MSE for Sigma = 1",xlab = "MODEL SIZE",ylab = "RMSE")
points(seq(1,9), test_rmse_mat[,1], col="red", pch=8)
lines(seq(1,9), test_rmse_mat[,1], col="red",lty=2)
legend("topright",legend=c("train rmse","test rmse"), col=c("blue","red"),pch=c("o","*"),lty=c(1,2), ncol=1)
The above plot shows the Train RMSE and Test RMSE for sigma = 1, is being very close to each other. As the model grow, both the Train / Test RMSE falls, however the tipping point is the Model 6, the test RMSE increases after the Model 6
plot(seq(1,9), train_rmse_mat[,2], type="o", col="blue", pch="o", lty=1,main="Train, Test MSE for Sigma = 2",xlab = "MODEL SIZE",ylab = "RMSE")
points(seq(1,9), test_rmse_mat[,2], col="red", pch=8)
lines(seq(1,9), test_rmse_mat[,2], col="red",lty=2)
legend("topright",legend=c("train rmse","test rmse"), col=c("blue","red"),pch=c("o","*"),lty=c(1,2), ncol=1)
The above plot shows the Train RMSE and Test RMSE for sigma = 2, is being very close to each other till model 6, after model 6 they are ditant apart to each other. Also As the model grow, both the Train / Test RMSE falls, however the tipping point is the Model 6, after Model 6, the Test RMSE increases
plot(seq(1,9), train_rmse_mat[,3], type="o", col="blue", pch="o", lty=1,main="Train, Test MSE for Sigma = 4",xlab = "MODEL SIZE",ylab = "RMSE")
points(seq(1,9), test_rmse_mat[,3], col="red", pch=8)
lines(seq(1,9), test_rmse_mat[,3], col="red",lty=2)
legend("topright",legend=c("train rmse","test rmse"), col=c("blue","red"),pch=c("o","*"),lty=c(1,2), ncol=1)
The above plot shows the Train RMSE and Test RMSE for sigma = 4, is being very close to each other till model 2, after model 2 they are distant apart to each other and distance between them increase as the model grows. Also as the model grow, both the Train / Test RMSE falls, however the tipping point is the Model 6, after the Model 6, the Test RMSE value increases
We will discuss on the following points, as we mentioned in the Introduction section
plot(test_rmse_model1_mat[,1],col="dodgerblue",ylim=(c(0.8,3)),pch=1,lty=1,xlab = "Index", ylab = "RMSE")
points(test_rmse_model2_mat[,1], col="orangered",ylim=(c(0.8,3)),pch=1,lty=2)
points(test_rmse_model3_mat[,1], col="blue4",ylim=(c(0.8,3)),pch=1,lty=3)
points(test_rmse_model4_mat[,1], col="limegreen",ylim=(c(0.8,3)),pch=1,lty=3)
points(test_rmse_model5_mat[,1], col="orange",ylim=(c(0.8,3)),pch=1,lty=4)
points(test_rmse_model6_mat[,1], col="darkorchid",ylim=(c(0.8,3)),pch=6,lty=5)
points(test_rmse_model7_mat[,1], col="violetred",ylim=(c(0.8,3)),pch=1,lty=6)
points(test_rmse_model8_mat[,1], col="tan3",ylim=(c(0.8,3)),pch=1,lty=7)
points(test_rmse_model9_mat[,1], col="deeppink",ylim=(c(0.8,3)),pch=1,lty=8)
legend("topright",legend=c("Model1","Model2","Model3","Model4","Model5","Model6","Model7","Model8","Model9"),col=c("dodgerblue","orangered","blue4","limegreen","orange","darkorchid","violetred","tan3","deeppink"),lty=c(1,2,3,4,5,6,7,8,9))
The above plots the RMSE of all model created from the simulation for the Sigma = 1
The correct model should be the one which should have the lowest RMSE. From the above plots (which plots against all the 1000 RMSE of simulation) it shows that the Model 1(blue color) and Model 2(red color) have higher RMSE and distinctly far from the other model (Model 3/4/5/6/7/8/9), lets ignore the Model 1 and Model 2. Let’s explore the other models as below
plot(test_rmse_model3_mat[,1], col="dodgerblue",ylim=(c(0.9,1.3)),pch=1,lty=3,xlab="Indx",ylab="RMSE")
points(test_rmse_model4_mat[,1], col="limegreen",ylim=(c(0.9,1.3)),pch=1,lty=3)
points(test_rmse_model5_mat[,1], col="orange",ylim=(c(0.9,1.3)),pch=1,lty=4)
points(test_rmse_model6_mat[,1], col="darkorchid",ylim=(c(0.9,1.3)),pch=8,lty=5)
points(test_rmse_model7_mat[,1], col="violetred",ylim=(c(0.9,1.3)),pch=1,lty=6)
points(test_rmse_model8_mat[,1], col="tan3",ylim=(c(0.9,1.3)),pch=1,lty=7)
points(test_rmse_model9_mat[,1], col="deeppink",ylim=(c(0.9,1.3)),pch=1,lty=8)
legend("topright",legend=c("Model3","Model4","Model5","Model6","Model7","Model8","Model9"),col=c("blue4","limegreen","orange","darkorchid","violetred","tan3","deeppink"),pch=c(1,1,1,8,1,1),lty=c(3,4,5,6,7,8,9))
The above model plots the RMSE of models (3/4/5/6/7/8/9) created from the simulation for the Sigma = 1
From the above plot, its clear that the Model 6 not always (marked in darkorchid) selected as the best model, because doesn’t shows the RMSE is lowest among the other models(2/3/4/5/7/8/9). There are many instances, where the Model 6 RMSE is higher than that of other models. Also though, there are instances where the Model 6 RMSE is the lowest among other models. However other models also have the RMSE in range with the model 6. Hence, if we consider all the 1000 simulated RMSE, Model 6 is not the best model always being selected. Let’s take a loot at the average plot of RMSE of all the models and plot
plot(seq(1,9), test_rmse_mat[,1], type="o", col="chartreuse4", pch=8, lty=1,main="Sigma = 1",xlab = "MODEL SIZE",ylab = "RMSE")
points(6, min(test_rmse_mat[,1]), col="chartreuse4",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 1
The above plot shows the model 6 has the lowest RMSE which is around 1.0148456, which runs on average RMSE for 1000 simulation over 9 models for the sigma = 1. In this methods (averaging) shows model 6 selects as the best model
plot(seq(1,9), test_rmse_mat[,2], type="o", col="blue4", pch="*", lty=1,main="Sigma = 2",xlab = "MODEL SIZE",ylab = "RMSE")
points(6, min(test_rmse_mat[,2]), col="blue4",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 2
The above plot shows the model 6 has the lowest RMSE which is around 2.030618, which runs on average RMSE for 1000 simulation over 9 models for the sigma = 1. In this methods (averaging) shows model 6 selects as the best model
plot(seq(1,9), test_rmse_mat[,3], type="o", col="red", pch="*", lty=1,main="Sigma = 4",xlab = "MODEL SIZE",ylab = "RMSE")
points(6, min(test_rmse_mat[,3]), col="red",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 4
The above plot shows the model 6 has the lowest RMSE which is around 4.0538772, which runs on average RMSE for 1000 simulation over 9 models for the sigma = 1. In this methods (averaging) shows model 6 selects as the best model
plot(seq(1,9), test_rmse_mat[,3], col="red",lty=1,ylim=c(0.5,5),xlab="MODEL SIZE",ylab="RMSE")
lines(seq(1,9), test_rmse_mat[,3], col="red",lty=1)
points(6, min(test_rmse_mat[,3]), col="red",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 4
points(seq(1,9), test_rmse_mat[,2], col="blue4",lty=2)
lines(seq(1,9), test_rmse_mat[,2], col="blue4",lty=2)
points(6, min(test_rmse_mat[,2]), col="blue4",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 2
points(seq(1,9), test_rmse_mat[,1], col="chartreuse4",lty=3)
lines(seq(1,9), test_rmse_mat[,1], col="chartreuse4",lty=3)
points(6, min(test_rmse_mat[,1]), col="chartreuse4",pch=16,cex=3) #This is to highlight the point where the RMSE is minimum for sigma = 1
legend("topright",legend=c("sigma = 4","sigma = 2","simga = 1"),col=c("red","blue4","chartreuse4"),lty=c(1,2,3))
The above plots shows the comparision of the average RMSE of all the 9 models for respective 3 sigma values (1, 2 and 4). The big filled circle points shows the lowest of the RMSE against the Model Size. From the comparision of the plot, it shows that lowest RMSE is from the sigma =1, also noted that higher the value of sigma, higher will be the Test RMSE. Hence to lower the RMSE for the test, we need to choose the lower value of noise (i.e. sigma)
In this simulation project we will investigate the power of the signifincance of the regression test for the simple linear regression (SLR) model
\(H_0 : \beta_{0} = 0 Vs H_1: \beta_{1} \neq 0\)
Where, Power is the probability of rejecting the null hypothesis when the null is not true, that is, the alternative is true and \(\beta_{1}\) is non-zero.
We will do the simulation of the Simple linear regression (SLR) of the model \[ Y_i = \beta_0 + \beta_1 x_{i1} + \epsilon_i.\].
For the simplicity we will make \(\beta_0\) = 0, thus \(\beta_1\) is essentially controlling the amount of signal. We will be considering the following for the different signals , noises and sample sizes
We will hold the value if the \(\alpha\) as 0.05, while validing the hypothesis. For the generation of the sample data, will be using the seq(0, 5, length = n) for different values of the sample size (i.e., n)
For each possible combination of \(\beta_{1}\) and \(\alpha\), we will simulate 1000 times to perform the significance of linear regression (SLR) and following formula will be using to measure the Power
\(\hat{Power}\) = \(\hat{P}\)[Reject \(H_0\), \(H_1\) True] = \(\dfrac{\#Test Rejected}{\#Simulations}\)
As part of this exercise, the following are the Goals to achieve
In order to achieve these goals, we have various methods in the form of R codes, markdown
Below are the methods proposed which will runs the 1000 simulations against β1, σ and n. After each simulations,
Once the matrix model_sim is populated, we will be using the matix to have different plots to address different discussion
birthday = 19770411
set.seed(birthday)
alpha = 0.05
beta_1 = seq(from=-2,to=2,by=0.1)
sigma = c(1,2,4)
n = c(10,20,30)
no_loop = 1000
#simple linear regression(SLR) Function to create sample data
slr_sim = function(n_in,sd=1,beta_1_in=-2){
x_values = seq(0, 5, length = n_in)
epsilon = rnorm(n_in,mean = 0,sd = sd)
y = beta_1_in * x_values + epsilon
data.frame(predictor = x_values,response=y)
}
#Matrix to store the Power of the simulated models
model_sim = cbind(beta_1_sim_val=rep(0,length(beta_1)),n_10_sigma_1=rep(0,length(beta_1)),n_20_sigma_1=rep(0,length(beta_1)),n_30_sigma_1=rep(0,length(beta_1)),n_10_sigma_2=rep(0,length(beta_1)),n_20_sigma_2=rep(0,length(beta_1)),n_30_sigma_2=rep(0,length(beta_1)),n_10_sigma_4=rep(0,length(beta_1)),n_20_sigma_4=rep(0,length(beta_1)),n_30_sigma_1=rep(0,length(beta_1)))
model_sim[,1] = seq(from=-2,to=2,by=0.1)
extra_counter = 0
#Loop to iterate over Sigma, number of samples, beta_1 values and 1000 simulations to extract and store the Power values
run_counter = 0
for(s in 1:length(sigma)){ #sigma = c(1,2,4)
for (n_s in 1:length(n)){ #n = c(10,20,30)
for (b in 1:length(beta_1)){ # beta_1 = seq(from=-2,to=2,by=0.1)
tot_sum_signifincace_sim = 0
run_counter = run_counter + 1
for(i in 1:no_loop){
slr_data = slr_sim(n_in = n[n_s], sd = sigma[s], beta_1_in = beta_1[b] )
model = lm(response~predictor, data = slr_data)
model_p_val = summary(model)$coefficient[2,4]
tot_sum_signifincace_sim = tot_sum_signifincace_sim + ifelse(model_p_val<alpha,1,0)
}
power_sim_beta = tot_sum_signifincace_sim / no_loop
model_sim[b,s+n_s+extra_counter] = power_sim_beta
print (paste(run_counter," Sigma:",sigma[s]," n:",n[n_s]," beta_1:",beta_1[b]," Signifiance_Sum:",tot_sum_signifincace_sim, " Power:",power_sim_beta))
}
}
extra_counter = extra_counter + 2
}
## [1] "1 Sigma: 1 n: 10 beta_1: -2 Signifiance_Sum: 1000 Power: 1"
## [1] "2 Sigma: 1 n: 10 beta_1: -1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "3 Sigma: 1 n: 10 beta_1: -1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "4 Sigma: 1 n: 10 beta_1: -1.7 Signifiance_Sum: 1000 Power: 1"
## [1] "5 Sigma: 1 n: 10 beta_1: -1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "6 Sigma: 1 n: 10 beta_1: -1.5 Signifiance_Sum: 1000 Power: 1"
## [1] "7 Sigma: 1 n: 10 beta_1: -1.4 Signifiance_Sum: 1000 Power: 1"
## [1] "8 Sigma: 1 n: 10 beta_1: -1.3 Signifiance_Sum: 1000 Power: 1"
## [1] "9 Sigma: 1 n: 10 beta_1: -1.2 Signifiance_Sum: 1000 Power: 1"
## [1] "10 Sigma: 1 n: 10 beta_1: -1.1 Signifiance_Sum: 998 Power: 0.998"
## [1] "11 Sigma: 1 n: 10 beta_1: -1 Signifiance_Sum: 992 Power: 0.992"
## [1] "12 Sigma: 1 n: 10 beta_1: -0.9 Signifiance_Sum: 980 Power: 0.98"
## [1] "13 Sigma: 1 n: 10 beta_1: -0.8 Signifiance_Sum: 935 Power: 0.935"
## [1] "14 Sigma: 1 n: 10 beta_1: -0.7 Signifiance_Sum: 880 Power: 0.88"
## [1] "15 Sigma: 1 n: 10 beta_1: -0.6 Signifiance_Sum: 743 Power: 0.743"
## [1] "16 Sigma: 1 n: 10 beta_1: -0.5 Signifiance_Sum: 599 Power: 0.599"
## [1] "17 Sigma: 1 n: 10 beta_1: -0.4 Signifiance_Sum: 430 Power: 0.43"
## [1] "18 Sigma: 1 n: 10 beta_1: -0.3 Signifiance_Sum: 270 Power: 0.27"
## [1] "19 Sigma: 1 n: 10 beta_1: -0.2 Signifiance_Sum: 130 Power: 0.13"
## [1] "20 Sigma: 1 n: 10 beta_1: -0.0999999999999999 Signifiance_Sum: 84 Power: 0.084"
## [1] "21 Sigma: 1 n: 10 beta_1: 0 Signifiance_Sum: 41 Power: 0.041"
## [1] "22 Sigma: 1 n: 10 beta_1: 0.1 Signifiance_Sum: 68 Power: 0.068"
## [1] "23 Sigma: 1 n: 10 beta_1: 0.2 Signifiance_Sum: 152 Power: 0.152"
## [1] "24 Sigma: 1 n: 10 beta_1: 0.3 Signifiance_Sum: 256 Power: 0.256"
## [1] "25 Sigma: 1 n: 10 beta_1: 0.4 Signifiance_Sum: 416 Power: 0.416"
## [1] "26 Sigma: 1 n: 10 beta_1: 0.5 Signifiance_Sum: 584 Power: 0.584"
## [1] "27 Sigma: 1 n: 10 beta_1: 0.6 Signifiance_Sum: 713 Power: 0.713"
## [1] "28 Sigma: 1 n: 10 beta_1: 0.7 Signifiance_Sum: 868 Power: 0.868"
## [1] "29 Sigma: 1 n: 10 beta_1: 0.8 Signifiance_Sum: 942 Power: 0.942"
## [1] "30 Sigma: 1 n: 10 beta_1: 0.9 Signifiance_Sum: 971 Power: 0.971"
## [1] "31 Sigma: 1 n: 10 beta_1: 1 Signifiance_Sum: 992 Power: 0.992"
## [1] "32 Sigma: 1 n: 10 beta_1: 1.1 Signifiance_Sum: 998 Power: 0.998"
## [1] "33 Sigma: 1 n: 10 beta_1: 1.2 Signifiance_Sum: 1000 Power: 1"
## [1] "34 Sigma: 1 n: 10 beta_1: 1.3 Signifiance_Sum: 1000 Power: 1"
## [1] "35 Sigma: 1 n: 10 beta_1: 1.4 Signifiance_Sum: 1000 Power: 1"
## [1] "36 Sigma: 1 n: 10 beta_1: 1.5 Signifiance_Sum: 1000 Power: 1"
## [1] "37 Sigma: 1 n: 10 beta_1: 1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "38 Sigma: 1 n: 10 beta_1: 1.7 Signifiance_Sum: 1000 Power: 1"
## [1] "39 Sigma: 1 n: 10 beta_1: 1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "40 Sigma: 1 n: 10 beta_1: 1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "41 Sigma: 1 n: 10 beta_1: 2 Signifiance_Sum: 1000 Power: 1"
## [1] "42 Sigma: 1 n: 20 beta_1: -2 Signifiance_Sum: 1000 Power: 1"
## [1] "43 Sigma: 1 n: 20 beta_1: -1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "44 Sigma: 1 n: 20 beta_1: -1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "45 Sigma: 1 n: 20 beta_1: -1.7 Signifiance_Sum: 1000 Power: 1"
## [1] "46 Sigma: 1 n: 20 beta_1: -1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "47 Sigma: 1 n: 20 beta_1: -1.5 Signifiance_Sum: 1000 Power: 1"
## [1] "48 Sigma: 1 n: 20 beta_1: -1.4 Signifiance_Sum: 1000 Power: 1"
## [1] "49 Sigma: 1 n: 20 beta_1: -1.3 Signifiance_Sum: 1000 Power: 1"
## [1] "50 Sigma: 1 n: 20 beta_1: -1.2 Signifiance_Sum: 1000 Power: 1"
## [1] "51 Sigma: 1 n: 20 beta_1: -1.1 Signifiance_Sum: 1000 Power: 1"
## [1] "52 Sigma: 1 n: 20 beta_1: -1 Signifiance_Sum: 1000 Power: 1"
## [1] "53 Sigma: 1 n: 20 beta_1: -0.9 Signifiance_Sum: 1000 Power: 1"
## [1] "54 Sigma: 1 n: 20 beta_1: -0.8 Signifiance_Sum: 1000 Power: 1"
## [1] "55 Sigma: 1 n: 20 beta_1: -0.7 Signifiance_Sum: 990 Power: 0.99"
## [1] "56 Sigma: 1 n: 20 beta_1: -0.6 Signifiance_Sum: 967 Power: 0.967"
## [1] "57 Sigma: 1 n: 20 beta_1: -0.5 Signifiance_Sum: 897 Power: 0.897"
## [1] "58 Sigma: 1 n: 20 beta_1: -0.4 Signifiance_Sum: 745 Power: 0.745"
## [1] "59 Sigma: 1 n: 20 beta_1: -0.3 Signifiance_Sum: 513 Power: 0.513"
## [1] "60 Sigma: 1 n: 20 beta_1: -0.2 Signifiance_Sum: 234 Power: 0.234"
## [1] "61 Sigma: 1 n: 20 beta_1: -0.0999999999999999 Signifiance_Sum: 116 Power: 0.116"
## [1] "62 Sigma: 1 n: 20 beta_1: 0 Signifiance_Sum: 40 Power: 0.04"
## [1] "63 Sigma: 1 n: 20 beta_1: 0.1 Signifiance_Sum: 108 Power: 0.108"
## [1] "64 Sigma: 1 n: 20 beta_1: 0.2 Signifiance_Sum: 237 Power: 0.237"
## [1] "65 Sigma: 1 n: 20 beta_1: 0.3 Signifiance_Sum: 500 Power: 0.5"
## [1] "66 Sigma: 1 n: 20 beta_1: 0.4 Signifiance_Sum: 727 Power: 0.727"
## [1] "67 Sigma: 1 n: 20 beta_1: 0.5 Signifiance_Sum: 908 Power: 0.908"
## [1] "68 Sigma: 1 n: 20 beta_1: 0.6 Signifiance_Sum: 966 Power: 0.966"
## [1] "69 Sigma: 1 n: 20 beta_1: 0.7 Signifiance_Sum: 999 Power: 0.999"
## [1] "70 Sigma: 1 n: 20 beta_1: 0.8 Signifiance_Sum: 998 Power: 0.998"
## [1] "71 Sigma: 1 n: 20 beta_1: 0.9 Signifiance_Sum: 1000 Power: 1"
## [1] "72 Sigma: 1 n: 20 beta_1: 1 Signifiance_Sum: 1000 Power: 1"
## [1] "73 Sigma: 1 n: 20 beta_1: 1.1 Signifiance_Sum: 1000 Power: 1"
## [1] "74 Sigma: 1 n: 20 beta_1: 1.2 Signifiance_Sum: 1000 Power: 1"
## [1] "75 Sigma: 1 n: 20 beta_1: 1.3 Signifiance_Sum: 1000 Power: 1"
## [1] "76 Sigma: 1 n: 20 beta_1: 1.4 Signifiance_Sum: 1000 Power: 1"
## [1] "77 Sigma: 1 n: 20 beta_1: 1.5 Signifiance_Sum: 1000 Power: 1"
## [1] "78 Sigma: 1 n: 20 beta_1: 1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "79 Sigma: 1 n: 20 beta_1: 1.7 Signifiance_Sum: 1000 Power: 1"
## [1] "80 Sigma: 1 n: 20 beta_1: 1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "81 Sigma: 1 n: 20 beta_1: 1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "82 Sigma: 1 n: 20 beta_1: 2 Signifiance_Sum: 1000 Power: 1"
## [1] "83 Sigma: 1 n: 30 beta_1: -2 Signifiance_Sum: 1000 Power: 1"
## [1] "84 Sigma: 1 n: 30 beta_1: -1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "85 Sigma: 1 n: 30 beta_1: -1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "86 Sigma: 1 n: 30 beta_1: -1.7 Signifiance_Sum: 1000 Power: 1"
## [1] "87 Sigma: 1 n: 30 beta_1: -1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "88 Sigma: 1 n: 30 beta_1: -1.5 Signifiance_Sum: 1000 Power: 1"
## [1] "89 Sigma: 1 n: 30 beta_1: -1.4 Signifiance_Sum: 1000 Power: 1"
## [1] "90 Sigma: 1 n: 30 beta_1: -1.3 Signifiance_Sum: 1000 Power: 1"
## [1] "91 Sigma: 1 n: 30 beta_1: -1.2 Signifiance_Sum: 1000 Power: 1"
## [1] "92 Sigma: 1 n: 30 beta_1: -1.1 Signifiance_Sum: 1000 Power: 1"
## [1] "93 Sigma: 1 n: 30 beta_1: -1 Signifiance_Sum: 1000 Power: 1"
## [1] "94 Sigma: 1 n: 30 beta_1: -0.9 Signifiance_Sum: 1000 Power: 1"
## [1] "95 Sigma: 1 n: 30 beta_1: -0.8 Signifiance_Sum: 1000 Power: 1"
## [1] "96 Sigma: 1 n: 30 beta_1: -0.7 Signifiance_Sum: 999 Power: 0.999"
## [1] "97 Sigma: 1 n: 30 beta_1: -0.6 Signifiance_Sum: 994 Power: 0.994"
## [1] "98 Sigma: 1 n: 30 beta_1: -0.5 Signifiance_Sum: 972 Power: 0.972"
## [1] "99 Sigma: 1 n: 30 beta_1: -0.4 Signifiance_Sum: 880 Power: 0.88"
## [1] "100 Sigma: 1 n: 30 beta_1: -0.3 Signifiance_Sum: 683 Power: 0.683"
## [1] "101 Sigma: 1 n: 30 beta_1: -0.2 Signifiance_Sum: 371 Power: 0.371"
## [1] "102 Sigma: 1 n: 30 beta_1: -0.0999999999999999 Signifiance_Sum: 130 Power: 0.13"
## [1] "103 Sigma: 1 n: 30 beta_1: 0 Signifiance_Sum: 41 Power: 0.041"
## [1] "104 Sigma: 1 n: 30 beta_1: 0.1 Signifiance_Sum: 127 Power: 0.127"
## [1] "105 Sigma: 1 n: 30 beta_1: 0.2 Signifiance_Sum: 345 Power: 0.345"
## [1] "106 Sigma: 1 n: 30 beta_1: 0.3 Signifiance_Sum: 664 Power: 0.664"
## [1] "107 Sigma: 1 n: 30 beta_1: 0.4 Signifiance_Sum: 889 Power: 0.889"
## [1] "108 Sigma: 1 n: 30 beta_1: 0.5 Signifiance_Sum: 983 Power: 0.983"
## [1] "109 Sigma: 1 n: 30 beta_1: 0.6 Signifiance_Sum: 996 Power: 0.996"
## [1] "110 Sigma: 1 n: 30 beta_1: 0.7 Signifiance_Sum: 1000 Power: 1"
## [1] "111 Sigma: 1 n: 30 beta_1: 0.8 Signifiance_Sum: 1000 Power: 1"
## [1] "112 Sigma: 1 n: 30 beta_1: 0.9 Signifiance_Sum: 1000 Power: 1"
## [1] "113 Sigma: 1 n: 30 beta_1: 1 Signifiance_Sum: 1000 Power: 1"
## [1] "114 Sigma: 1 n: 30 beta_1: 1.1 Signifiance_Sum: 1000 Power: 1"
## [1] "115 Sigma: 1 n: 30 beta_1: 1.2 Signifiance_Sum: 1000 Power: 1"
## [1] "116 Sigma: 1 n: 30 beta_1: 1.3 Signifiance_Sum: 1000 Power: 1"
## [1] "117 Sigma: 1 n: 30 beta_1: 1.4 Signifiance_Sum: 1000 Power: 1"
## [1] "118 Sigma: 1 n: 30 beta_1: 1.5 Signifiance_Sum: 1000 Power: 1"
## [1] "119 Sigma: 1 n: 30 beta_1: 1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "120 Sigma: 1 n: 30 beta_1: 1.7 Signifiance_Sum: 1000 Power: 1"
## [1] "121 Sigma: 1 n: 30 beta_1: 1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "122 Sigma: 1 n: 30 beta_1: 1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "123 Sigma: 1 n: 30 beta_1: 2 Signifiance_Sum: 1000 Power: 1"
## [1] "124 Sigma: 2 n: 10 beta_1: -2 Signifiance_Sum: 991 Power: 0.991"
## [1] "125 Sigma: 2 n: 10 beta_1: -1.9 Signifiance_Sum: 990 Power: 0.99"
## [1] "126 Sigma: 2 n: 10 beta_1: -1.8 Signifiance_Sum: 975 Power: 0.975"
## [1] "127 Sigma: 2 n: 10 beta_1: -1.7 Signifiance_Sum: 957 Power: 0.957"
## [1] "128 Sigma: 2 n: 10 beta_1: -1.6 Signifiance_Sum: 942 Power: 0.942"
## [1] "129 Sigma: 2 n: 10 beta_1: -1.5 Signifiance_Sum: 906 Power: 0.906"
## [1] "130 Sigma: 2 n: 10 beta_1: -1.4 Signifiance_Sum: 887 Power: 0.887"
## [1] "131 Sigma: 2 n: 10 beta_1: -1.3 Signifiance_Sum: 831 Power: 0.831"
## [1] "132 Sigma: 2 n: 10 beta_1: -1.2 Signifiance_Sum: 742 Power: 0.742"
## [1] "133 Sigma: 2 n: 10 beta_1: -1.1 Signifiance_Sum: 676 Power: 0.676"
## [1] "134 Sigma: 2 n: 10 beta_1: -1 Signifiance_Sum: 587 Power: 0.587"
## [1] "135 Sigma: 2 n: 10 beta_1: -0.9 Signifiance_Sum: 562 Power: 0.562"
## [1] "136 Sigma: 2 n: 10 beta_1: -0.8 Signifiance_Sum: 404 Power: 0.404"
## [1] "137 Sigma: 2 n: 10 beta_1: -0.7 Signifiance_Sum: 347 Power: 0.347"
## [1] "138 Sigma: 2 n: 10 beta_1: -0.6 Signifiance_Sum: 263 Power: 0.263"
## [1] "139 Sigma: 2 n: 10 beta_1: -0.5 Signifiance_Sum: 199 Power: 0.199"
## [1] "140 Sigma: 2 n: 10 beta_1: -0.4 Signifiance_Sum: 140 Power: 0.14"
## [1] "141 Sigma: 2 n: 10 beta_1: -0.3 Signifiance_Sum: 108 Power: 0.108"
## [1] "142 Sigma: 2 n: 10 beta_1: -0.2 Signifiance_Sum: 63 Power: 0.063"
## [1] "143 Sigma: 2 n: 10 beta_1: -0.0999999999999999 Signifiance_Sum: 56 Power: 0.056"
## [1] "144 Sigma: 2 n: 10 beta_1: 0 Signifiance_Sum: 44 Power: 0.044"
## [1] "145 Sigma: 2 n: 10 beta_1: 0.1 Signifiance_Sum: 60 Power: 0.06"
## [1] "146 Sigma: 2 n: 10 beta_1: 0.2 Signifiance_Sum: 77 Power: 0.077"
## [1] "147 Sigma: 2 n: 10 beta_1: 0.3 Signifiance_Sum: 98 Power: 0.098"
## [1] "148 Sigma: 2 n: 10 beta_1: 0.4 Signifiance_Sum: 158 Power: 0.158"
## [1] "149 Sigma: 2 n: 10 beta_1: 0.5 Signifiance_Sum: 214 Power: 0.214"
## [1] "150 Sigma: 2 n: 10 beta_1: 0.6 Signifiance_Sum: 258 Power: 0.258"
## [1] "151 Sigma: 2 n: 10 beta_1: 0.7 Signifiance_Sum: 338 Power: 0.338"
## [1] "152 Sigma: 2 n: 10 beta_1: 0.8 Signifiance_Sum: 424 Power: 0.424"
## [1] "153 Sigma: 2 n: 10 beta_1: 0.9 Signifiance_Sum: 518 Power: 0.518"
## [1] "154 Sigma: 2 n: 10 beta_1: 1 Signifiance_Sum: 605 Power: 0.605"
## [1] "155 Sigma: 2 n: 10 beta_1: 1.1 Signifiance_Sum: 689 Power: 0.689"
## [1] "156 Sigma: 2 n: 10 beta_1: 1.2 Signifiance_Sum: 772 Power: 0.772"
## [1] "157 Sigma: 2 n: 10 beta_1: 1.3 Signifiance_Sum: 829 Power: 0.829"
## [1] "158 Sigma: 2 n: 10 beta_1: 1.4 Signifiance_Sum: 887 Power: 0.887"
## [1] "159 Sigma: 2 n: 10 beta_1: 1.5 Signifiance_Sum: 922 Power: 0.922"
## [1] "160 Sigma: 2 n: 10 beta_1: 1.6 Signifiance_Sum: 948 Power: 0.948"
## [1] "161 Sigma: 2 n: 10 beta_1: 1.7 Signifiance_Sum: 969 Power: 0.969"
## [1] "162 Sigma: 2 n: 10 beta_1: 1.8 Signifiance_Sum: 974 Power: 0.974"
## [1] "163 Sigma: 2 n: 10 beta_1: 1.9 Signifiance_Sum: 983 Power: 0.983"
## [1] "164 Sigma: 2 n: 10 beta_1: 2 Signifiance_Sum: 996 Power: 0.996"
## [1] "165 Sigma: 2 n: 20 beta_1: -2 Signifiance_Sum: 1000 Power: 1"
## [1] "166 Sigma: 2 n: 20 beta_1: -1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "167 Sigma: 2 n: 20 beta_1: -1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "168 Sigma: 2 n: 20 beta_1: -1.7 Signifiance_Sum: 999 Power: 0.999"
## [1] "169 Sigma: 2 n: 20 beta_1: -1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "170 Sigma: 2 n: 20 beta_1: -1.5 Signifiance_Sum: 999 Power: 0.999"
## [1] "171 Sigma: 2 n: 20 beta_1: -1.4 Signifiance_Sum: 992 Power: 0.992"
## [1] "172 Sigma: 2 n: 20 beta_1: -1.3 Signifiance_Sum: 984 Power: 0.984"
## [1] "173 Sigma: 2 n: 20 beta_1: -1.2 Signifiance_Sum: 967 Power: 0.967"
## [1] "174 Sigma: 2 n: 20 beta_1: -1.1 Signifiance_Sum: 939 Power: 0.939"
## [1] "175 Sigma: 2 n: 20 beta_1: -1 Signifiance_Sum: 890 Power: 0.89"
## [1] "176 Sigma: 2 n: 20 beta_1: -0.9 Signifiance_Sum: 828 Power: 0.828"
## [1] "177 Sigma: 2 n: 20 beta_1: -0.8 Signifiance_Sum: 738 Power: 0.738"
## [1] "178 Sigma: 2 n: 20 beta_1: -0.7 Signifiance_Sum: 601 Power: 0.601"
## [1] "179 Sigma: 2 n: 20 beta_1: -0.6 Signifiance_Sum: 489 Power: 0.489"
## [1] "180 Sigma: 2 n: 20 beta_1: -0.5 Signifiance_Sum: 343 Power: 0.343"
## [1] "181 Sigma: 2 n: 20 beta_1: -0.4 Signifiance_Sum: 254 Power: 0.254"
## [1] "182 Sigma: 2 n: 20 beta_1: -0.3 Signifiance_Sum: 145 Power: 0.145"
## [1] "183 Sigma: 2 n: 20 beta_1: -0.2 Signifiance_Sum: 106 Power: 0.106"
## [1] "184 Sigma: 2 n: 20 beta_1: -0.0999999999999999 Signifiance_Sum: 64 Power: 0.064"
## [1] "185 Sigma: 2 n: 20 beta_1: 0 Signifiance_Sum: 45 Power: 0.045"
## [1] "186 Sigma: 2 n: 20 beta_1: 0.1 Signifiance_Sum: 62 Power: 0.062"
## [1] "187 Sigma: 2 n: 20 beta_1: 0.2 Signifiance_Sum: 105 Power: 0.105"
## [1] "188 Sigma: 2 n: 20 beta_1: 0.3 Signifiance_Sum: 160 Power: 0.16"
## [1] "189 Sigma: 2 n: 20 beta_1: 0.4 Signifiance_Sum: 232 Power: 0.232"
## [1] "190 Sigma: 2 n: 20 beta_1: 0.5 Signifiance_Sum: 366 Power: 0.366"
## [1] "191 Sigma: 2 n: 20 beta_1: 0.6 Signifiance_Sum: 461 Power: 0.461"
## [1] "192 Sigma: 2 n: 20 beta_1: 0.7 Signifiance_Sum: 602 Power: 0.602"
## [1] "193 Sigma: 2 n: 20 beta_1: 0.8 Signifiance_Sum: 716 Power: 0.716"
## [1] "194 Sigma: 2 n: 20 beta_1: 0.9 Signifiance_Sum: 822 Power: 0.822"
## [1] "195 Sigma: 2 n: 20 beta_1: 1 Signifiance_Sum: 904 Power: 0.904"
## [1] "196 Sigma: 2 n: 20 beta_1: 1.1 Signifiance_Sum: 948 Power: 0.948"
## [1] "197 Sigma: 2 n: 20 beta_1: 1.2 Signifiance_Sum: 968 Power: 0.968"
## [1] "198 Sigma: 2 n: 20 beta_1: 1.3 Signifiance_Sum: 981 Power: 0.981"
## [1] "199 Sigma: 2 n: 20 beta_1: 1.4 Signifiance_Sum: 994 Power: 0.994"
## [1] "200 Sigma: 2 n: 20 beta_1: 1.5 Signifiance_Sum: 997 Power: 0.997"
## [1] "201 Sigma: 2 n: 20 beta_1: 1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "202 Sigma: 2 n: 20 beta_1: 1.7 Signifiance_Sum: 1000 Power: 1"
## [1] "203 Sigma: 2 n: 20 beta_1: 1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "204 Sigma: 2 n: 20 beta_1: 1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "205 Sigma: 2 n: 20 beta_1: 2 Signifiance_Sum: 1000 Power: 1"
## [1] "206 Sigma: 2 n: 30 beta_1: -2 Signifiance_Sum: 1000 Power: 1"
## [1] "207 Sigma: 2 n: 30 beta_1: -1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "208 Sigma: 2 n: 30 beta_1: -1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "209 Sigma: 2 n: 30 beta_1: -1.7 Signifiance_Sum: 1000 Power: 1"
## [1] "210 Sigma: 2 n: 30 beta_1: -1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "211 Sigma: 2 n: 30 beta_1: -1.5 Signifiance_Sum: 1000 Power: 1"
## [1] "212 Sigma: 2 n: 30 beta_1: -1.4 Signifiance_Sum: 1000 Power: 1"
## [1] "213 Sigma: 2 n: 30 beta_1: -1.3 Signifiance_Sum: 1000 Power: 1"
## [1] "214 Sigma: 2 n: 30 beta_1: -1.2 Signifiance_Sum: 998 Power: 0.998"
## [1] "215 Sigma: 2 n: 30 beta_1: -1.1 Signifiance_Sum: 990 Power: 0.99"
## [1] "216 Sigma: 2 n: 30 beta_1: -1 Signifiance_Sum: 973 Power: 0.973"
## [1] "217 Sigma: 2 n: 30 beta_1: -0.9 Signifiance_Sum: 932 Power: 0.932"
## [1] "218 Sigma: 2 n: 30 beta_1: -0.8 Signifiance_Sum: 890 Power: 0.89"
## [1] "219 Sigma: 2 n: 30 beta_1: -0.7 Signifiance_Sum: 790 Power: 0.79"
## [1] "220 Sigma: 2 n: 30 beta_1: -0.6 Signifiance_Sum: 653 Power: 0.653"
## [1] "221 Sigma: 2 n: 30 beta_1: -0.5 Signifiance_Sum: 526 Power: 0.526"
## [1] "222 Sigma: 2 n: 30 beta_1: -0.4 Signifiance_Sum: 343 Power: 0.343"
## [1] "223 Sigma: 2 n: 30 beta_1: -0.3 Signifiance_Sum: 219 Power: 0.219"
## [1] "224 Sigma: 2 n: 30 beta_1: -0.2 Signifiance_Sum: 125 Power: 0.125"
## [1] "225 Sigma: 2 n: 30 beta_1: -0.0999999999999999 Signifiance_Sum: 53 Power: 0.053"
## [1] "226 Sigma: 2 n: 30 beta_1: 0 Signifiance_Sum: 54 Power: 0.054"
## [1] "227 Sigma: 2 n: 30 beta_1: 0.1 Signifiance_Sum: 67 Power: 0.067"
## [1] "228 Sigma: 2 n: 30 beta_1: 0.2 Signifiance_Sum: 142 Power: 0.142"
## [1] "229 Sigma: 2 n: 30 beta_1: 0.3 Signifiance_Sum: 226 Power: 0.226"
## [1] "230 Sigma: 2 n: 30 beta_1: 0.4 Signifiance_Sum: 345 Power: 0.345"
## [1] "231 Sigma: 2 n: 30 beta_1: 0.5 Signifiance_Sum: 497 Power: 0.497"
## [1] "232 Sigma: 2 n: 30 beta_1: 0.6 Signifiance_Sum: 687 Power: 0.687"
## [1] "233 Sigma: 2 n: 30 beta_1: 0.7 Signifiance_Sum: 784 Power: 0.784"
## [1] "234 Sigma: 2 n: 30 beta_1: 0.8 Signifiance_Sum: 888 Power: 0.888"
## [1] "235 Sigma: 2 n: 30 beta_1: 0.9 Signifiance_Sum: 931 Power: 0.931"
## [1] "236 Sigma: 2 n: 30 beta_1: 1 Signifiance_Sum: 988 Power: 0.988"
## [1] "237 Sigma: 2 n: 30 beta_1: 1.1 Signifiance_Sum: 989 Power: 0.989"
## [1] "238 Sigma: 2 n: 30 beta_1: 1.2 Signifiance_Sum: 997 Power: 0.997"
## [1] "239 Sigma: 2 n: 30 beta_1: 1.3 Signifiance_Sum: 999 Power: 0.999"
## [1] "240 Sigma: 2 n: 30 beta_1: 1.4 Signifiance_Sum: 1000 Power: 1"
## [1] "241 Sigma: 2 n: 30 beta_1: 1.5 Signifiance_Sum: 1000 Power: 1"
## [1] "242 Sigma: 2 n: 30 beta_1: 1.6 Signifiance_Sum: 1000 Power: 1"
## [1] "243 Sigma: 2 n: 30 beta_1: 1.7 Signifiance_Sum: 1000 Power: 1"
## [1] "244 Sigma: 2 n: 30 beta_1: 1.8 Signifiance_Sum: 1000 Power: 1"
## [1] "245 Sigma: 2 n: 30 beta_1: 1.9 Signifiance_Sum: 1000 Power: 1"
## [1] "246 Sigma: 2 n: 30 beta_1: 2 Signifiance_Sum: 1000 Power: 1"
## [1] "247 Sigma: 4 n: 10 beta_1: -2 Signifiance_Sum: 578 Power: 0.578"
## [1] "248 Sigma: 4 n: 10 beta_1: -1.9 Signifiance_Sum: 551 Power: 0.551"
## [1] "249 Sigma: 4 n: 10 beta_1: -1.8 Signifiance_Sum: 520 Power: 0.52"
## [1] "250 Sigma: 4 n: 10 beta_1: -1.7 Signifiance_Sum: 485 Power: 0.485"
## [1] "251 Sigma: 4 n: 10 beta_1: -1.6 Signifiance_Sum: 448 Power: 0.448"
## [1] "252 Sigma: 4 n: 10 beta_1: -1.5 Signifiance_Sum: 390 Power: 0.39"
## [1] "253 Sigma: 4 n: 10 beta_1: -1.4 Signifiance_Sum: 373 Power: 0.373"
## [1] "254 Sigma: 4 n: 10 beta_1: -1.3 Signifiance_Sum: 288 Power: 0.288"
## [1] "255 Sigma: 4 n: 10 beta_1: -1.2 Signifiance_Sum: 284 Power: 0.284"
## [1] "256 Sigma: 4 n: 10 beta_1: -1.1 Signifiance_Sum: 223 Power: 0.223"
## [1] "257 Sigma: 4 n: 10 beta_1: -1 Signifiance_Sum: 196 Power: 0.196"
## [1] "258 Sigma: 4 n: 10 beta_1: -0.9 Signifiance_Sum: 178 Power: 0.178"
## [1] "259 Sigma: 4 n: 10 beta_1: -0.8 Signifiance_Sum: 146 Power: 0.146"
## [1] "260 Sigma: 4 n: 10 beta_1: -0.7 Signifiance_Sum: 110 Power: 0.11"
## [1] "261 Sigma: 4 n: 10 beta_1: -0.6 Signifiance_Sum: 105 Power: 0.105"
## [1] "262 Sigma: 4 n: 10 beta_1: -0.5 Signifiance_Sum: 75 Power: 0.075"
## [1] "263 Sigma: 4 n: 10 beta_1: -0.4 Signifiance_Sum: 66 Power: 0.066"
## [1] "264 Sigma: 4 n: 10 beta_1: -0.3 Signifiance_Sum: 56 Power: 0.056"
## [1] "265 Sigma: 4 n: 10 beta_1: -0.2 Signifiance_Sum: 52 Power: 0.052"
## [1] "266 Sigma: 4 n: 10 beta_1: -0.0999999999999999 Signifiance_Sum: 45 Power: 0.045"
## [1] "267 Sigma: 4 n: 10 beta_1: 0 Signifiance_Sum: 44 Power: 0.044"
## [1] "268 Sigma: 4 n: 10 beta_1: 0.1 Signifiance_Sum: 58 Power: 0.058"
## [1] "269 Sigma: 4 n: 10 beta_1: 0.2 Signifiance_Sum: 54 Power: 0.054"
## [1] "270 Sigma: 4 n: 10 beta_1: 0.3 Signifiance_Sum: 74 Power: 0.074"
## [1] "271 Sigma: 4 n: 10 beta_1: 0.4 Signifiance_Sum: 60 Power: 0.06"
## [1] "272 Sigma: 4 n: 10 beta_1: 0.5 Signifiance_Sum: 70 Power: 0.07"
## [1] "273 Sigma: 4 n: 10 beta_1: 0.6 Signifiance_Sum: 107 Power: 0.107"
## [1] "274 Sigma: 4 n: 10 beta_1: 0.7 Signifiance_Sum: 107 Power: 0.107"
## [1] "275 Sigma: 4 n: 10 beta_1: 0.8 Signifiance_Sum: 141 Power: 0.141"
## [1] "276 Sigma: 4 n: 10 beta_1: 0.9 Signifiance_Sum: 201 Power: 0.201"
## [1] "277 Sigma: 4 n: 10 beta_1: 1 Signifiance_Sum: 190 Power: 0.19"
## [1] "278 Sigma: 4 n: 10 beta_1: 1.1 Signifiance_Sum: 248 Power: 0.248"
## [1] "279 Sigma: 4 n: 10 beta_1: 1.2 Signifiance_Sum: 258 Power: 0.258"
## [1] "280 Sigma: 4 n: 10 beta_1: 1.3 Signifiance_Sum: 314 Power: 0.314"
## [1] "281 Sigma: 4 n: 10 beta_1: 1.4 Signifiance_Sum: 354 Power: 0.354"
## [1] "282 Sigma: 4 n: 10 beta_1: 1.5 Signifiance_Sum: 381 Power: 0.381"
## [1] "283 Sigma: 4 n: 10 beta_1: 1.6 Signifiance_Sum: 453 Power: 0.453"
## [1] "284 Sigma: 4 n: 10 beta_1: 1.7 Signifiance_Sum: 464 Power: 0.464"
## [1] "285 Sigma: 4 n: 10 beta_1: 1.8 Signifiance_Sum: 504 Power: 0.504"
## [1] "286 Sigma: 4 n: 10 beta_1: 1.9 Signifiance_Sum: 545 Power: 0.545"
## [1] "287 Sigma: 4 n: 10 beta_1: 2 Signifiance_Sum: 591 Power: 0.591"
## [1] "288 Sigma: 4 n: 20 beta_1: -2 Signifiance_Sum: 889 Power: 0.889"
## [1] "289 Sigma: 4 n: 20 beta_1: -1.9 Signifiance_Sum: 873 Power: 0.873"
## [1] "290 Sigma: 4 n: 20 beta_1: -1.8 Signifiance_Sum: 827 Power: 0.827"
## [1] "291 Sigma: 4 n: 20 beta_1: -1.7 Signifiance_Sum: 773 Power: 0.773"
## [1] "292 Sigma: 4 n: 20 beta_1: -1.6 Signifiance_Sum: 726 Power: 0.726"
## [1] "293 Sigma: 4 n: 20 beta_1: -1.5 Signifiance_Sum: 671 Power: 0.671"
## [1] "294 Sigma: 4 n: 20 beta_1: -1.4 Signifiance_Sum: 618 Power: 0.618"
## [1] "295 Sigma: 4 n: 20 beta_1: -1.3 Signifiance_Sum: 553 Power: 0.553"
## [1] "296 Sigma: 4 n: 20 beta_1: -1.2 Signifiance_Sum: 480 Power: 0.48"
## [1] "297 Sigma: 4 n: 20 beta_1: -1.1 Signifiance_Sum: 364 Power: 0.364"
## [1] "298 Sigma: 4 n: 20 beta_1: -1 Signifiance_Sum: 347 Power: 0.347"
## [1] "299 Sigma: 4 n: 20 beta_1: -0.9 Signifiance_Sum: 301 Power: 0.301"
## [1] "300 Sigma: 4 n: 20 beta_1: -0.8 Signifiance_Sum: 248 Power: 0.248"
## [1] "301 Sigma: 4 n: 20 beta_1: -0.7 Signifiance_Sum: 195 Power: 0.195"
## [1] "302 Sigma: 4 n: 20 beta_1: -0.6 Signifiance_Sum: 155 Power: 0.155"
## [1] "303 Sigma: 4 n: 20 beta_1: -0.5 Signifiance_Sum: 118 Power: 0.118"
## [1] "304 Sigma: 4 n: 20 beta_1: -0.4 Signifiance_Sum: 111 Power: 0.111"
## [1] "305 Sigma: 4 n: 20 beta_1: -0.3 Signifiance_Sum: 68 Power: 0.068"
## [1] "306 Sigma: 4 n: 20 beta_1: -0.2 Signifiance_Sum: 53 Power: 0.053"
## [1] "307 Sigma: 4 n: 20 beta_1: -0.0999999999999999 Signifiance_Sum: 46 Power: 0.046"
## [1] "308 Sigma: 4 n: 20 beta_1: 0 Signifiance_Sum: 40 Power: 0.04"
## [1] "309 Sigma: 4 n: 20 beta_1: 0.1 Signifiance_Sum: 47 Power: 0.047"
## [1] "310 Sigma: 4 n: 20 beta_1: 0.2 Signifiance_Sum: 64 Power: 0.064"
## [1] "311 Sigma: 4 n: 20 beta_1: 0.3 Signifiance_Sum: 73 Power: 0.073"
## [1] "312 Sigma: 4 n: 20 beta_1: 0.4 Signifiance_Sum: 100 Power: 0.1"
## [1] "313 Sigma: 4 n: 20 beta_1: 0.5 Signifiance_Sum: 123 Power: 0.123"
## [1] "314 Sigma: 4 n: 20 beta_1: 0.6 Signifiance_Sum: 144 Power: 0.144"
## [1] "315 Sigma: 4 n: 20 beta_1: 0.7 Signifiance_Sum: 194 Power: 0.194"
## [1] "316 Sigma: 4 n: 20 beta_1: 0.8 Signifiance_Sum: 235 Power: 0.235"
## [1] "317 Sigma: 4 n: 20 beta_1: 0.9 Signifiance_Sum: 334 Power: 0.334"
## [1] "318 Sigma: 4 n: 20 beta_1: 1 Signifiance_Sum: 382 Power: 0.382"
## [1] "319 Sigma: 4 n: 20 beta_1: 1.1 Signifiance_Sum: 410 Power: 0.41"
## [1] "320 Sigma: 4 n: 20 beta_1: 1.2 Signifiance_Sum: 461 Power: 0.461"
## [1] "321 Sigma: 4 n: 20 beta_1: 1.3 Signifiance_Sum: 523 Power: 0.523"
## [1] "322 Sigma: 4 n: 20 beta_1: 1.4 Signifiance_Sum: 614 Power: 0.614"
## [1] "323 Sigma: 4 n: 20 beta_1: 1.5 Signifiance_Sum: 669 Power: 0.669"
## [1] "324 Sigma: 4 n: 20 beta_1: 1.6 Signifiance_Sum: 724 Power: 0.724"
## [1] "325 Sigma: 4 n: 20 beta_1: 1.7 Signifiance_Sum: 788 Power: 0.788"
## [1] "326 Sigma: 4 n: 20 beta_1: 1.8 Signifiance_Sum: 841 Power: 0.841"
## [1] "327 Sigma: 4 n: 20 beta_1: 1.9 Signifiance_Sum: 852 Power: 0.852"
## [1] "328 Sigma: 4 n: 20 beta_1: 2 Signifiance_Sum: 910 Power: 0.91"
## [1] "329 Sigma: 4 n: 30 beta_1: -2 Signifiance_Sum: 978 Power: 0.978"
## [1] "330 Sigma: 4 n: 30 beta_1: -1.9 Signifiance_Sum: 964 Power: 0.964"
## [1] "331 Sigma: 4 n: 30 beta_1: -1.8 Signifiance_Sum: 945 Power: 0.945"
## [1] "332 Sigma: 4 n: 30 beta_1: -1.7 Signifiance_Sum: 923 Power: 0.923"
## [1] "333 Sigma: 4 n: 30 beta_1: -1.6 Signifiance_Sum: 855 Power: 0.855"
## [1] "334 Sigma: 4 n: 30 beta_1: -1.5 Signifiance_Sum: 851 Power: 0.851"
## [1] "335 Sigma: 4 n: 30 beta_1: -1.4 Signifiance_Sum: 802 Power: 0.802"
## [1] "336 Sigma: 4 n: 30 beta_1: -1.3 Signifiance_Sum: 731 Power: 0.731"
## [1] "337 Sigma: 4 n: 30 beta_1: -1.2 Signifiance_Sum: 649 Power: 0.649"
## [1] "338 Sigma: 4 n: 30 beta_1: -1.1 Signifiance_Sum: 565 Power: 0.565"
## [1] "339 Sigma: 4 n: 30 beta_1: -1 Signifiance_Sum: 510 Power: 0.51"
## [1] "340 Sigma: 4 n: 30 beta_1: -0.9 Signifiance_Sum: 453 Power: 0.453"
## [1] "341 Sigma: 4 n: 30 beta_1: -0.8 Signifiance_Sum: 352 Power: 0.352"
## [1] "342 Sigma: 4 n: 30 beta_1: -0.7 Signifiance_Sum: 282 Power: 0.282"
## [1] "343 Sigma: 4 n: 30 beta_1: -0.6 Signifiance_Sum: 222 Power: 0.222"
## [1] "344 Sigma: 4 n: 30 beta_1: -0.5 Signifiance_Sum: 178 Power: 0.178"
## [1] "345 Sigma: 4 n: 30 beta_1: -0.4 Signifiance_Sum: 123 Power: 0.123"
## [1] "346 Sigma: 4 n: 30 beta_1: -0.3 Signifiance_Sum: 87 Power: 0.087"
## [1] "347 Sigma: 4 n: 30 beta_1: -0.2 Signifiance_Sum: 65 Power: 0.065"
## [1] "348 Sigma: 4 n: 30 beta_1: -0.0999999999999999 Signifiance_Sum: 56 Power: 0.056"
## [1] "349 Sigma: 4 n: 30 beta_1: 0 Signifiance_Sum: 56 Power: 0.056"
## [1] "350 Sigma: 4 n: 30 beta_1: 0.1 Signifiance_Sum: 52 Power: 0.052"
## [1] "351 Sigma: 4 n: 30 beta_1: 0.2 Signifiance_Sum: 67 Power: 0.067"
## [1] "352 Sigma: 4 n: 30 beta_1: 0.3 Signifiance_Sum: 95 Power: 0.095"
## [1] "353 Sigma: 4 n: 30 beta_1: 0.4 Signifiance_Sum: 118 Power: 0.118"
## [1] "354 Sigma: 4 n: 30 beta_1: 0.5 Signifiance_Sum: 165 Power: 0.165"
## [1] "355 Sigma: 4 n: 30 beta_1: 0.6 Signifiance_Sum: 236 Power: 0.236"
## [1] "356 Sigma: 4 n: 30 beta_1: 0.7 Signifiance_Sum: 282 Power: 0.282"
## [1] "357 Sigma: 4 n: 30 beta_1: 0.8 Signifiance_Sum: 370 Power: 0.37"
## [1] "358 Sigma: 4 n: 30 beta_1: 0.9 Signifiance_Sum: 411 Power: 0.411"
## [1] "359 Sigma: 4 n: 30 beta_1: 1 Signifiance_Sum: 501 Power: 0.501"
## [1] "360 Sigma: 4 n: 30 beta_1: 1.1 Signifiance_Sum: 572 Power: 0.572"
## [1] "361 Sigma: 4 n: 30 beta_1: 1.2 Signifiance_Sum: 669 Power: 0.669"
## [1] "362 Sigma: 4 n: 30 beta_1: 1.3 Signifiance_Sum: 746 Power: 0.746"
## [1] "363 Sigma: 4 n: 30 beta_1: 1.4 Signifiance_Sum: 783 Power: 0.783"
## [1] "364 Sigma: 4 n: 30 beta_1: 1.5 Signifiance_Sum: 845 Power: 0.845"
## [1] "365 Sigma: 4 n: 30 beta_1: 1.6 Signifiance_Sum: 863 Power: 0.863"
## [1] "366 Sigma: 4 n: 30 beta_1: 1.7 Signifiance_Sum: 915 Power: 0.915"
## [1] "367 Sigma: 4 n: 30 beta_1: 1.8 Signifiance_Sum: 940 Power: 0.94"
## [1] "368 Sigma: 4 n: 30 beta_1: 1.9 Signifiance_Sum: 965 Power: 0.965"
## [1] "369 Sigma: 4 n: 30 beta_1: 2 Signifiance_Sum: 974 Power: 0.974"
As we have discussed in the Introduction on the aspect of Goal, we have to producded plots to support these Goals
#Below code plot the effect of Signal (β1) against the Power for noise of sigma = 1 against all sample size (10,20 and 30)
plot(model_sim[,1],model_sim[,2], type="o", col="dodgerblue", pch=1, lty=1,main="Sigma = 1",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="red", pch=8,lty=2)
lines(model_sim[,1], model_sim[,3], col="red",lty=2)
points(model_sim[,1], model_sim[,4], col="darkorchid", pch=16,lty=3)
lines(model_sim[,1], model_sim[,4], col="darkorchid",lty=3)
legend("topright",legend=c("n = 10","n = 20"," n = 30"),col=c("dodgerblue","red","darkorchid"),pch=c(1,8,16),lty=c(1,2,3))
The above plot depicts the following for sigma = 1
plot(model_sim[,1],model_sim[,5], type="o", col="dodgerblue", pch=1, lty=1,main="Sigma = 2",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,6], col="red", pch=8,lty=2)
lines(model_sim[,1], model_sim[,6], col="red",lty=2)
points(model_sim[,1], model_sim[,7], col="darkorchid", pch=16,lty=3)
lines(model_sim[,1], model_sim[,7], col="darkorchid",lty=3)
legend("topright",legend=c("n = 10","n = 20"," n = 30"),col=c("dodgerblue","red","darkorchid"),pch=c(1,8,16),lty=c(1,2,3))
The above plot depicts the following for the sigma = 2
plot(model_sim[,1],model_sim[,8], type="o", col="dodgerblue", pch=1, lty=1,main="Sigma = 4",xlab = "beta_1",ylab = "Power",ylim=c(0,1))
points(model_sim[,1], model_sim[,9], col="red", pch=8,lty=2)
lines(model_sim[,1], model_sim[,9], col="red",lty=2)
points(model_sim[,1], model_sim[,10], col="darkorchid", pch=16,lty=3)
lines(model_sim[,1], model_sim[,10], col="darkorchid",lty=3)
legend("topright",legend=c("n = 10","n = 20"," n = 30"),col=c("dodgerblue","red","darkorchid"),pch=c(1,8,16),lty=c(1,2,3))
The above plot depicts the following for tsigma = 4
Now as part of the goal, let’s discuss the what’s the impact of β1, n and σ with respect to power. We will be plotting different plots to conclude the summary of the impact
plot(model_sim[,1],model_sim[,2], type="o", col="dodgerblue", pch=1, lty=1,main="Effect of sigma (1,2,4) on the Power",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="dodgerblue", pch=1,lty=1)
lines(model_sim[,1], model_sim[,3], col="dodgerblue",lty=1)
points(model_sim[,1], model_sim[,4], col="dodgerblue", pch=1,lty=1)
lines(model_sim[,1], model_sim[,4], col="dodgerblue",lty=1)
points(model_sim[,1],model_sim[,5], type="o", col="green", pch=1, lty=2)
lines(model_sim[,1],model_sim[,5], type="o", col="green", pch=1, lty=2)
points(model_sim[,1], model_sim[,6], col="green", pch=1,lty=2)
lines(model_sim[,1], model_sim[,6], col="green",lty=2)
points(model_sim[,1], model_sim[,7], col="green", pch=1,lty=2)
lines(model_sim[,1], model_sim[,7], col="green",lty=2)
points(model_sim[,1],model_sim[,8], type="o", col="red", pch=1, lty=3)
lines(model_sim[,1],model_sim[,8], type="o", col="red", pch=1, lty=3)
points(model_sim[,1], model_sim[,9], col="red", pch=1,lty=3)
lines(model_sim[,1], model_sim[,9], col="red",lty=3)
points(model_sim[,1], model_sim[,10], col="red", pch=1,lty=3)
lines(model_sim[,1], model_sim[,10], col="red",lty=3)
legend("topright",legend=c("sigma=1,n=10,20,30","sigma=2,n=10,20,30","sigma=3,n=10,20,30"),col=c("dodgerblue","green","darkorchid","red"),pch=c(1,1,1),lty=c(1,2,3))
Below are the observations on the effect of σ on power
par(mfrow=c(2, 2))
plot(model_sim[,1],model_sim[,2], type="o", col="deeppink", pch=1, lty=1,main="Sigma=1",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="darkolivegreen4", pch=1,lty=2)
lines(model_sim[,1], model_sim[,3], col="darkolivegreen4",lty=2)
points(model_sim[,1], model_sim[,4], col="blueviolet", pch=1,lty=3)
lines(model_sim[,1], model_sim[,4], col="blueviolet",lty=3)
legend("topright",legend=c("n=10","n=20","n=30"),col=c("deeppink","darkolivegreen4","blueviolet"),pch=c(1,1,1),lty=c(1,2,3))
plot(model_sim[,1],model_sim[,5], type="o", col="deeppink", pch=1, lty=1,main="Sigma=2",xlab = "beta_1",ylab = "Power")
#points(model_sim[,1],model_sim[,5], type="o", col="deeppink", pch=1, lty=1)
lines(model_sim[,1],model_sim[,5], type="o", col="deeppink", pch=1, lty=1)
points(model_sim[,1], model_sim[,6], col="darkolivegreen4", pch=1,lty=2)
lines(model_sim[,1], model_sim[,6], col="darkolivegreen4",lty=2)
points(model_sim[,1], model_sim[,7], col="blueviolet", pch=1,lty=3)
lines(model_sim[,1], model_sim[,7], col="blueviolet",lty=3)
legend("topright",legend=c("n=10","n=20","n=30"),col=c("deeppink","darkolivegreen4","blueviolet"),pch=c(1,1,1),lty=c(1,2,3))
plot(model_sim[,1],model_sim[,8], type="o", col="deeppink", pch=1, lty=1,main="Sigma=4",xlab = "beta_1",ylab = "Power",ylim=c(0,1))
#points(model_sim[,1],model_sim[,8], type="o", col="deeppink", pch=1, lty=1)
lines(model_sim[,1],model_sim[,8], type="o", col="deeppink", pch=1, lty=1)
points(model_sim[,1], model_sim[,9], col="darkolivegreen4", pch=1,lty=2)
lines(model_sim[,1], model_sim[,9], col="darkolivegreen4",lty=2)
points(model_sim[,1], model_sim[,10], col="blueviolet", pch=1,lty=3)
lines(model_sim[,1], model_sim[,10], col="blueviolet",lty=3)
legend("topright",legend=c("n=10","n=20","n=30"),col=c("deeppink","darkolivegreen4","blueviolet"),pch=c(1,1,1),lty=c(1,2,3))
Below are the observations on the effect of sample size (n = 10,20 and 30) on the beta_1
plot(model_sim[,1],model_sim[,2], type="o", col="dodgerblue", pch=1, lty=1,main="Sigma = 1,2,4 & n = 10,20,30",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="red", pch=2,lty=2)
lines(model_sim[,1], model_sim[,3], col="red",lty=2)
points(model_sim[,1], model_sim[,4], col="tan1", pch=3,lty=3)
lines(model_sim[,1], model_sim[,4], col="tan1",lty=3)
points(model_sim[,1],model_sim[,5], type="o", col="yellow", pch=4, lty=4)
lines(model_sim[,1],model_sim[,5], type="o", col="yellow", pch=4, lty=4)
points(model_sim[,1], model_sim[,6], col="deeppink", pch=5,lty=5)
lines(model_sim[,1], model_sim[,6], col="deeppink",lty=5)
points(model_sim[,1], model_sim[,7], col="darkolivegreen4", pch=6,lty=6)
lines(model_sim[,1], model_sim[,7], col="darkolivegreen4",lty=6)
points(model_sim[,1],model_sim[,8], type="o", col="darkgreen", pch=7, lty=7)
lines(model_sim[,1],model_sim[,8], type="o", col="darkgreen", pch=7, lty=7)
points(model_sim[,1], model_sim[,9], col="darkblue", pch=8,lty=8)
lines(model_sim[,1], model_sim[,9], col="darkblue",lty=8)
points(model_sim[,1], model_sim[,10], col="blueviolet", pch=9,lty=9)
lines(model_sim[,1], model_sim[,10], col="blueviolet",lty=9)
legend("topright",legend=c("sigma=1,n=10","sigma=1,n=20","sigma=1,n=30","sigma=2,n=10","sigma=2,n=20","sigma=2,n=30","sigma=3,n=10","sigma=3,n=20","sigma=3,n=30"),col=c("dodgerblue","red","darkorchid","yellow","deeppink","darkolivegreen4","darkgreen","darkblue","blueviolet"),pch=c(1,2,3,4,5,6,7,8,9),lty=c(1,2,3,4,5,6,7,8,9))
If we combine all the plots for all the sample size beta(β1), sample size(n) and sigma(σ), from the above figure, below are the observations on the effect of β1 on power *
Let’s simulate the MLR for 3000 iteration and record the Train / Test RMSE to see what’s the change
birthday = 19770411
set.seed(birthday)
no_loop = 3000
#Matrix to store the Power of the simulated models
model_sim_3000 = cbind(beta_1_sim_val=rep(0,length(beta_1)),n_10_sigma_1=rep(0,length(beta_1)),n_20_sigma_1=rep(0,length(beta_1)),n_30_sigma_1=rep(0,length(beta_1)),n_10_sigma_2=rep(0,length(beta_1)),n_20_sigma_2=rep(0,length(beta_1)),n_30_sigma_2=rep(0,length(beta_1)),n_10_sigma_4=rep(0,length(beta_1)),n_20_sigma_4=rep(0,length(beta_1)),n_30_sigma_1=rep(0,length(beta_1)))
model_sim_3000[,1] = seq(from=-2,to=2,by=0.1)
extra_counter = 0
#Loop to iterate over Sigma, number of samples, beta_1 values and 1000 simulations to extract and store the Power values
run_counter = 0
for(s in 1:length(sigma)){ #sigma = c(1,2,4)
for (n_s in 1:length(n)){ #n = c(10,20,30)
for (b in 1:length(beta_1)){ # beta_1 = seq(from=-2,to=2,by=0.1)
tot_sum_signifincace_sim = 0
run_counter = run_counter + 1
for(i in 1:no_loop){
slr_data = slr_sim(n_in = n[n_s], sd = sigma[s], beta_1_in = beta_1[b] )
model = lm(response~predictor, data = slr_data)
model_p_val = summary(model)$coefficient[2,4]
tot_sum_signifincace_sim = tot_sum_signifincace_sim + ifelse(model_p_val<alpha,1,0)
}
power_sim_beta = tot_sum_signifincace_sim / no_loop
model_sim_3000[b,s+n_s+extra_counter] = power_sim_beta
print (paste(run_counter," Sigma:",sigma[s]," n:",n[n_s]," beta_1:",beta_1[b]," Signifiance_Sum:",tot_sum_signifincace_sim, " Power:",power_sim_beta))
}
}
extra_counter = extra_counter + 2
}
## [1] "1 Sigma: 1 n: 10 beta_1: -2 Signifiance_Sum: 3000 Power: 1"
## [1] "2 Sigma: 1 n: 10 beta_1: -1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "3 Sigma: 1 n: 10 beta_1: -1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "4 Sigma: 1 n: 10 beta_1: -1.7 Signifiance_Sum: 3000 Power: 1"
## [1] "5 Sigma: 1 n: 10 beta_1: -1.6 Signifiance_Sum: 3000 Power: 1"
## [1] "6 Sigma: 1 n: 10 beta_1: -1.5 Signifiance_Sum: 3000 Power: 1"
## [1] "7 Sigma: 1 n: 10 beta_1: -1.4 Signifiance_Sum: 3000 Power: 1"
## [1] "8 Sigma: 1 n: 10 beta_1: -1.3 Signifiance_Sum: 3000 Power: 1"
## [1] "9 Sigma: 1 n: 10 beta_1: -1.2 Signifiance_Sum: 2997 Power: 0.999"
## [1] "10 Sigma: 1 n: 10 beta_1: -1.1 Signifiance_Sum: 2993 Power: 0.997666666666667"
## [1] "11 Sigma: 1 n: 10 beta_1: -1 Signifiance_Sum: 2978 Power: 0.992666666666667"
## [1] "12 Sigma: 1 n: 10 beta_1: -0.9 Signifiance_Sum: 2938 Power: 0.979333333333333"
## [1] "13 Sigma: 1 n: 10 beta_1: -0.8 Signifiance_Sum: 2821 Power: 0.940333333333333"
## [1] "14 Sigma: 1 n: 10 beta_1: -0.7 Signifiance_Sum: 2610 Power: 0.87"
## [1] "15 Sigma: 1 n: 10 beta_1: -0.6 Signifiance_Sum: 2287 Power: 0.762333333333333"
## [1] "16 Sigma: 1 n: 10 beta_1: -0.5 Signifiance_Sum: 1839 Power: 0.613"
## [1] "17 Sigma: 1 n: 10 beta_1: -0.4 Signifiance_Sum: 1277 Power: 0.425666666666667"
## [1] "18 Sigma: 1 n: 10 beta_1: -0.3 Signifiance_Sum: 765 Power: 0.255"
## [1] "19 Sigma: 1 n: 10 beta_1: -0.2 Signifiance_Sum: 438 Power: 0.146"
## [1] "20 Sigma: 1 n: 10 beta_1: -0.0999999999999999 Signifiance_Sum: 231 Power: 0.077"
## [1] "21 Sigma: 1 n: 10 beta_1: 0 Signifiance_Sum: 160 Power: 0.0533333333333333"
## [1] "22 Sigma: 1 n: 10 beta_1: 0.1 Signifiance_Sum: 227 Power: 0.0756666666666667"
## [1] "23 Sigma: 1 n: 10 beta_1: 0.2 Signifiance_Sum: 468 Power: 0.156"
## [1] "24 Sigma: 1 n: 10 beta_1: 0.3 Signifiance_Sum: 812 Power: 0.270666666666667"
## [1] "25 Sigma: 1 n: 10 beta_1: 0.4 Signifiance_Sum: 1277 Power: 0.425666666666667"
## [1] "26 Sigma: 1 n: 10 beta_1: 0.5 Signifiance_Sum: 1829 Power: 0.609666666666667"
## [1] "27 Sigma: 1 n: 10 beta_1: 0.6 Signifiance_Sum: 2239 Power: 0.746333333333333"
## [1] "28 Sigma: 1 n: 10 beta_1: 0.7 Signifiance_Sum: 2638 Power: 0.879333333333333"
## [1] "29 Sigma: 1 n: 10 beta_1: 0.8 Signifiance_Sum: 2798 Power: 0.932666666666667"
## [1] "30 Sigma: 1 n: 10 beta_1: 0.9 Signifiance_Sum: 2934 Power: 0.978"
## [1] "31 Sigma: 1 n: 10 beta_1: 1 Signifiance_Sum: 2981 Power: 0.993666666666667"
## [1] "32 Sigma: 1 n: 10 beta_1: 1.1 Signifiance_Sum: 2993 Power: 0.997666666666667"
## [1] "33 Sigma: 1 n: 10 beta_1: 1.2 Signifiance_Sum: 2997 Power: 0.999"
## [1] "34 Sigma: 1 n: 10 beta_1: 1.3 Signifiance_Sum: 3000 Power: 1"
## [1] "35 Sigma: 1 n: 10 beta_1: 1.4 Signifiance_Sum: 3000 Power: 1"
## [1] "36 Sigma: 1 n: 10 beta_1: 1.5 Signifiance_Sum: 3000 Power: 1"
## [1] "37 Sigma: 1 n: 10 beta_1: 1.6 Signifiance_Sum: 3000 Power: 1"
## [1] "38 Sigma: 1 n: 10 beta_1: 1.7 Signifiance_Sum: 3000 Power: 1"
## [1] "39 Sigma: 1 n: 10 beta_1: 1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "40 Sigma: 1 n: 10 beta_1: 1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "41 Sigma: 1 n: 10 beta_1: 2 Signifiance_Sum: 3000 Power: 1"
## [1] "42 Sigma: 1 n: 20 beta_1: -2 Signifiance_Sum: 3000 Power: 1"
## [1] "43 Sigma: 1 n: 20 beta_1: -1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "44 Sigma: 1 n: 20 beta_1: -1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "45 Sigma: 1 n: 20 beta_1: -1.7 Signifiance_Sum: 3000 Power: 1"
## [1] "46 Sigma: 1 n: 20 beta_1: -1.6 Signifiance_Sum: 3000 Power: 1"
## [1] "47 Sigma: 1 n: 20 beta_1: -1.5 Signifiance_Sum: 3000 Power: 1"
## [1] "48 Sigma: 1 n: 20 beta_1: -1.4 Signifiance_Sum: 3000 Power: 1"
## [1] "49 Sigma: 1 n: 20 beta_1: -1.3 Signifiance_Sum: 3000 Power: 1"
## [1] "50 Sigma: 1 n: 20 beta_1: -1.2 Signifiance_Sum: 3000 Power: 1"
## [1] "51 Sigma: 1 n: 20 beta_1: -1.1 Signifiance_Sum: 3000 Power: 1"
## [1] "52 Sigma: 1 n: 20 beta_1: -1 Signifiance_Sum: 3000 Power: 1"
## [1] "53 Sigma: 1 n: 20 beta_1: -0.9 Signifiance_Sum: 2999 Power: 0.999666666666667"
## [1] "54 Sigma: 1 n: 20 beta_1: -0.8 Signifiance_Sum: 2999 Power: 0.999666666666667"
## [1] "55 Sigma: 1 n: 20 beta_1: -0.7 Signifiance_Sum: 2986 Power: 0.995333333333333"
## [1] "56 Sigma: 1 n: 20 beta_1: -0.6 Signifiance_Sum: 2933 Power: 0.977666666666667"
## [1] "57 Sigma: 1 n: 20 beta_1: -0.5 Signifiance_Sum: 2663 Power: 0.887666666666667"
## [1] "58 Sigma: 1 n: 20 beta_1: -0.4 Signifiance_Sum: 2199 Power: 0.733"
## [1] "59 Sigma: 1 n: 20 beta_1: -0.3 Signifiance_Sum: 1498 Power: 0.499333333333333"
## [1] "60 Sigma: 1 n: 20 beta_1: -0.2 Signifiance_Sum: 730 Power: 0.243333333333333"
## [1] "61 Sigma: 1 n: 20 beta_1: -0.0999999999999999 Signifiance_Sum: 303 Power: 0.101"
## [1] "62 Sigma: 1 n: 20 beta_1: 0 Signifiance_Sum: 157 Power: 0.0523333333333333"
## [1] "63 Sigma: 1 n: 20 beta_1: 0.1 Signifiance_Sum: 301 Power: 0.100333333333333"
## [1] "64 Sigma: 1 n: 20 beta_1: 0.2 Signifiance_Sum: 704 Power: 0.234666666666667"
## [1] "65 Sigma: 1 n: 20 beta_1: 0.3 Signifiance_Sum: 1458 Power: 0.486"
## [1] "66 Sigma: 1 n: 20 beta_1: 0.4 Signifiance_Sum: 2182 Power: 0.727333333333333"
## [1] "67 Sigma: 1 n: 20 beta_1: 0.5 Signifiance_Sum: 2648 Power: 0.882666666666667"
## [1] "68 Sigma: 1 n: 20 beta_1: 0.6 Signifiance_Sum: 2915 Power: 0.971666666666667"
## [1] "69 Sigma: 1 n: 20 beta_1: 0.7 Signifiance_Sum: 2984 Power: 0.994666666666667"
## [1] "70 Sigma: 1 n: 20 beta_1: 0.8 Signifiance_Sum: 2998 Power: 0.999333333333333"
## [1] "71 Sigma: 1 n: 20 beta_1: 0.9 Signifiance_Sum: 3000 Power: 1"
## [1] "72 Sigma: 1 n: 20 beta_1: 1 Signifiance_Sum: 3000 Power: 1"
## [1] "73 Sigma: 1 n: 20 beta_1: 1.1 Signifiance_Sum: 3000 Power: 1"
## [1] "74 Sigma: 1 n: 20 beta_1: 1.2 Signifiance_Sum: 3000 Power: 1"
## [1] "75 Sigma: 1 n: 20 beta_1: 1.3 Signifiance_Sum: 3000 Power: 1"
## [1] "76 Sigma: 1 n: 20 beta_1: 1.4 Signifiance_Sum: 3000 Power: 1"
## [1] "77 Sigma: 1 n: 20 beta_1: 1.5 Signifiance_Sum: 3000 Power: 1"
## [1] "78 Sigma: 1 n: 20 beta_1: 1.6 Signifiance_Sum: 3000 Power: 1"
## [1] "79 Sigma: 1 n: 20 beta_1: 1.7 Signifiance_Sum: 3000 Power: 1"
## [1] "80 Sigma: 1 n: 20 beta_1: 1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "81 Sigma: 1 n: 20 beta_1: 1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "82 Sigma: 1 n: 20 beta_1: 2 Signifiance_Sum: 3000 Power: 1"
## [1] "83 Sigma: 1 n: 30 beta_1: -2 Signifiance_Sum: 3000 Power: 1"
## [1] "84 Sigma: 1 n: 30 beta_1: -1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "85 Sigma: 1 n: 30 beta_1: -1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "86 Sigma: 1 n: 30 beta_1: -1.7 Signifiance_Sum: 3000 Power: 1"
## [1] "87 Sigma: 1 n: 30 beta_1: -1.6 Signifiance_Sum: 3000 Power: 1"
## [1] "88 Sigma: 1 n: 30 beta_1: -1.5 Signifiance_Sum: 3000 Power: 1"
## [1] "89 Sigma: 1 n: 30 beta_1: -1.4 Signifiance_Sum: 3000 Power: 1"
## [1] "90 Sigma: 1 n: 30 beta_1: -1.3 Signifiance_Sum: 3000 Power: 1"
## [1] "91 Sigma: 1 n: 30 beta_1: -1.2 Signifiance_Sum: 3000 Power: 1"
## [1] "92 Sigma: 1 n: 30 beta_1: -1.1 Signifiance_Sum: 3000 Power: 1"
## [1] "93 Sigma: 1 n: 30 beta_1: -1 Signifiance_Sum: 3000 Power: 1"
## [1] "94 Sigma: 1 n: 30 beta_1: -0.9 Signifiance_Sum: 3000 Power: 1"
## [1] "95 Sigma: 1 n: 30 beta_1: -0.8 Signifiance_Sum: 3000 Power: 1"
## [1] "96 Sigma: 1 n: 30 beta_1: -0.7 Signifiance_Sum: 2999 Power: 0.999666666666667"
## [1] "97 Sigma: 1 n: 30 beta_1: -0.6 Signifiance_Sum: 2993 Power: 0.997666666666667"
## [1] "98 Sigma: 1 n: 30 beta_1: -0.5 Signifiance_Sum: 2916 Power: 0.972"
## [1] "99 Sigma: 1 n: 30 beta_1: -0.4 Signifiance_Sum: 2654 Power: 0.884666666666667"
## [1] "100 Sigma: 1 n: 30 beta_1: -0.3 Signifiance_Sum: 1949 Power: 0.649666666666667"
## [1] "101 Sigma: 1 n: 30 beta_1: -0.2 Signifiance_Sum: 1007 Power: 0.335666666666667"
## [1] "102 Sigma: 1 n: 30 beta_1: -0.0999999999999999 Signifiance_Sum: 374 Power: 0.124666666666667"
## [1] "103 Sigma: 1 n: 30 beta_1: 0 Signifiance_Sum: 152 Power: 0.0506666666666667"
## [1] "104 Sigma: 1 n: 30 beta_1: 0.1 Signifiance_Sum: 344 Power: 0.114666666666667"
## [1] "105 Sigma: 1 n: 30 beta_1: 0.2 Signifiance_Sum: 1078 Power: 0.359333333333333"
## [1] "106 Sigma: 1 n: 30 beta_1: 0.3 Signifiance_Sum: 1940 Power: 0.646666666666667"
## [1] "107 Sigma: 1 n: 30 beta_1: 0.4 Signifiance_Sum: 2683 Power: 0.894333333333333"
## [1] "108 Sigma: 1 n: 30 beta_1: 0.5 Signifiance_Sum: 2939 Power: 0.979666666666667"
## [1] "109 Sigma: 1 n: 30 beta_1: 0.6 Signifiance_Sum: 2987 Power: 0.995666666666667"
## [1] "110 Sigma: 1 n: 30 beta_1: 0.7 Signifiance_Sum: 3000 Power: 1"
## [1] "111 Sigma: 1 n: 30 beta_1: 0.8 Signifiance_Sum: 3000 Power: 1"
## [1] "112 Sigma: 1 n: 30 beta_1: 0.9 Signifiance_Sum: 3000 Power: 1"
## [1] "113 Sigma: 1 n: 30 beta_1: 1 Signifiance_Sum: 3000 Power: 1"
## [1] "114 Sigma: 1 n: 30 beta_1: 1.1 Signifiance_Sum: 3000 Power: 1"
## [1] "115 Sigma: 1 n: 30 beta_1: 1.2 Signifiance_Sum: 3000 Power: 1"
## [1] "116 Sigma: 1 n: 30 beta_1: 1.3 Signifiance_Sum: 3000 Power: 1"
## [1] "117 Sigma: 1 n: 30 beta_1: 1.4 Signifiance_Sum: 3000 Power: 1"
## [1] "118 Sigma: 1 n: 30 beta_1: 1.5 Signifiance_Sum: 3000 Power: 1"
## [1] "119 Sigma: 1 n: 30 beta_1: 1.6 Signifiance_Sum: 3000 Power: 1"
## [1] "120 Sigma: 1 n: 30 beta_1: 1.7 Signifiance_Sum: 3000 Power: 1"
## [1] "121 Sigma: 1 n: 30 beta_1: 1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "122 Sigma: 1 n: 30 beta_1: 1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "123 Sigma: 1 n: 30 beta_1: 2 Signifiance_Sum: 3000 Power: 1"
## [1] "124 Sigma: 2 n: 10 beta_1: -2 Signifiance_Sum: 2972 Power: 0.990666666666667"
## [1] "125 Sigma: 2 n: 10 beta_1: -1.9 Signifiance_Sum: 2955 Power: 0.985"
## [1] "126 Sigma: 2 n: 10 beta_1: -1.8 Signifiance_Sum: 2922 Power: 0.974"
## [1] "127 Sigma: 2 n: 10 beta_1: -1.7 Signifiance_Sum: 2878 Power: 0.959333333333333"
## [1] "128 Sigma: 2 n: 10 beta_1: -1.6 Signifiance_Sum: 2845 Power: 0.948333333333333"
## [1] "129 Sigma: 2 n: 10 beta_1: -1.5 Signifiance_Sum: 2737 Power: 0.912333333333333"
## [1] "130 Sigma: 2 n: 10 beta_1: -1.4 Signifiance_Sum: 2614 Power: 0.871333333333333"
## [1] "131 Sigma: 2 n: 10 beta_1: -1.3 Signifiance_Sum: 2457 Power: 0.819"
## [1] "132 Sigma: 2 n: 10 beta_1: -1.2 Signifiance_Sum: 2296 Power: 0.765333333333333"
## [1] "133 Sigma: 2 n: 10 beta_1: -1.1 Signifiance_Sum: 2027 Power: 0.675666666666667"
## [1] "134 Sigma: 2 n: 10 beta_1: -1 Signifiance_Sum: 1818 Power: 0.606"
## [1] "135 Sigma: 2 n: 10 beta_1: -0.9 Signifiance_Sum: 1583 Power: 0.527666666666667"
## [1] "136 Sigma: 2 n: 10 beta_1: -0.8 Signifiance_Sum: 1299 Power: 0.433"
## [1] "137 Sigma: 2 n: 10 beta_1: -0.7 Signifiance_Sum: 1031 Power: 0.343666666666667"
## [1] "138 Sigma: 2 n: 10 beta_1: -0.6 Signifiance_Sum: 811 Power: 0.270333333333333"
## [1] "139 Sigma: 2 n: 10 beta_1: -0.5 Signifiance_Sum: 601 Power: 0.200333333333333"
## [1] "140 Sigma: 2 n: 10 beta_1: -0.4 Signifiance_Sum: 420 Power: 0.14"
## [1] "141 Sigma: 2 n: 10 beta_1: -0.3 Signifiance_Sum: 300 Power: 0.1"
## [1] "142 Sigma: 2 n: 10 beta_1: -0.2 Signifiance_Sum: 209 Power: 0.0696666666666667"
## [1] "143 Sigma: 2 n: 10 beta_1: -0.0999999999999999 Signifiance_Sum: 156 Power: 0.052"
## [1] "144 Sigma: 2 n: 10 beta_1: 0 Signifiance_Sum: 154 Power: 0.0513333333333333"
## [1] "145 Sigma: 2 n: 10 beta_1: 0.1 Signifiance_Sum: 180 Power: 0.06"
## [1] "146 Sigma: 2 n: 10 beta_1: 0.2 Signifiance_Sum: 209 Power: 0.0696666666666667"
## [1] "147 Sigma: 2 n: 10 beta_1: 0.3 Signifiance_Sum: 332 Power: 0.110666666666667"
## [1] "148 Sigma: 2 n: 10 beta_1: 0.4 Signifiance_Sum: 471 Power: 0.157"
## [1] "149 Sigma: 2 n: 10 beta_1: 0.5 Signifiance_Sum: 577 Power: 0.192333333333333"
## [1] "150 Sigma: 2 n: 10 beta_1: 0.6 Signifiance_Sum: 809 Power: 0.269666666666667"
## [1] "151 Sigma: 2 n: 10 beta_1: 0.7 Signifiance_Sum: 1046 Power: 0.348666666666667"
## [1] "152 Sigma: 2 n: 10 beta_1: 0.8 Signifiance_Sum: 1280 Power: 0.426666666666667"
## [1] "153 Sigma: 2 n: 10 beta_1: 0.9 Signifiance_Sum: 1534 Power: 0.511333333333333"
## [1] "154 Sigma: 2 n: 10 beta_1: 1 Signifiance_Sum: 1773 Power: 0.591"
## [1] "155 Sigma: 2 n: 10 beta_1: 1.1 Signifiance_Sum: 2078 Power: 0.692666666666667"
## [1] "156 Sigma: 2 n: 10 beta_1: 1.2 Signifiance_Sum: 2266 Power: 0.755333333333333"
## [1] "157 Sigma: 2 n: 10 beta_1: 1.3 Signifiance_Sum: 2445 Power: 0.815"
## [1] "158 Sigma: 2 n: 10 beta_1: 1.4 Signifiance_Sum: 2652 Power: 0.884"
## [1] "159 Sigma: 2 n: 10 beta_1: 1.5 Signifiance_Sum: 2733 Power: 0.911"
## [1] "160 Sigma: 2 n: 10 beta_1: 1.6 Signifiance_Sum: 2823 Power: 0.941"
## [1] "161 Sigma: 2 n: 10 beta_1: 1.7 Signifiance_Sum: 2895 Power: 0.965"
## [1] "162 Sigma: 2 n: 10 beta_1: 1.8 Signifiance_Sum: 2931 Power: 0.977"
## [1] "163 Sigma: 2 n: 10 beta_1: 1.9 Signifiance_Sum: 2961 Power: 0.987"
## [1] "164 Sigma: 2 n: 10 beta_1: 2 Signifiance_Sum: 2978 Power: 0.992666666666667"
## [1] "165 Sigma: 2 n: 20 beta_1: -2 Signifiance_Sum: 2999 Power: 0.999666666666667"
## [1] "166 Sigma: 2 n: 20 beta_1: -1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "167 Sigma: 2 n: 20 beta_1: -1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "168 Sigma: 2 n: 20 beta_1: -1.7 Signifiance_Sum: 2999 Power: 0.999666666666667"
## [1] "169 Sigma: 2 n: 20 beta_1: -1.6 Signifiance_Sum: 2996 Power: 0.998666666666667"
## [1] "170 Sigma: 2 n: 20 beta_1: -1.5 Signifiance_Sum: 2986 Power: 0.995333333333333"
## [1] "171 Sigma: 2 n: 20 beta_1: -1.4 Signifiance_Sum: 2985 Power: 0.995"
## [1] "172 Sigma: 2 n: 20 beta_1: -1.3 Signifiance_Sum: 2974 Power: 0.991333333333333"
## [1] "173 Sigma: 2 n: 20 beta_1: -1.2 Signifiance_Sum: 2904 Power: 0.968"
## [1] "174 Sigma: 2 n: 20 beta_1: -1.1 Signifiance_Sum: 2824 Power: 0.941333333333333"
## [1] "175 Sigma: 2 n: 20 beta_1: -1 Signifiance_Sum: 2683 Power: 0.894333333333333"
## [1] "176 Sigma: 2 n: 20 beta_1: -0.9 Signifiance_Sum: 2487 Power: 0.829"
## [1] "177 Sigma: 2 n: 20 beta_1: -0.8 Signifiance_Sum: 2202 Power: 0.734"
## [1] "178 Sigma: 2 n: 20 beta_1: -0.7 Signifiance_Sum: 1840 Power: 0.613333333333333"
## [1] "179 Sigma: 2 n: 20 beta_1: -0.6 Signifiance_Sum: 1453 Power: 0.484333333333333"
## [1] "180 Sigma: 2 n: 20 beta_1: -0.5 Signifiance_Sum: 1088 Power: 0.362666666666667"
## [1] "181 Sigma: 2 n: 20 beta_1: -0.4 Signifiance_Sum: 756 Power: 0.252"
## [1] "182 Sigma: 2 n: 20 beta_1: -0.3 Signifiance_Sum: 531 Power: 0.177"
## [1] "183 Sigma: 2 n: 20 beta_1: -0.2 Signifiance_Sum: 305 Power: 0.101666666666667"
## [1] "184 Sigma: 2 n: 20 beta_1: -0.0999999999999999 Signifiance_Sum: 184 Power: 0.0613333333333333"
## [1] "185 Sigma: 2 n: 20 beta_1: 0 Signifiance_Sum: 130 Power: 0.0433333333333333"
## [1] "186 Sigma: 2 n: 20 beta_1: 0.1 Signifiance_Sum: 202 Power: 0.0673333333333333"
## [1] "187 Sigma: 2 n: 20 beta_1: 0.2 Signifiance_Sum: 299 Power: 0.0996666666666667"
## [1] "188 Sigma: 2 n: 20 beta_1: 0.3 Signifiance_Sum: 482 Power: 0.160666666666667"
## [1] "189 Sigma: 2 n: 20 beta_1: 0.4 Signifiance_Sum: 746 Power: 0.248666666666667"
## [1] "190 Sigma: 2 n: 20 beta_1: 0.5 Signifiance_Sum: 1050 Power: 0.35"
## [1] "191 Sigma: 2 n: 20 beta_1: 0.6 Signifiance_Sum: 1457 Power: 0.485666666666667"
## [1] "192 Sigma: 2 n: 20 beta_1: 0.7 Signifiance_Sum: 1854 Power: 0.618"
## [1] "193 Sigma: 2 n: 20 beta_1: 0.8 Signifiance_Sum: 2156 Power: 0.718666666666667"
## [1] "194 Sigma: 2 n: 20 beta_1: 0.9 Signifiance_Sum: 2521 Power: 0.840333333333333"
## [1] "195 Sigma: 2 n: 20 beta_1: 1 Signifiance_Sum: 2704 Power: 0.901333333333333"
## [1] "196 Sigma: 2 n: 20 beta_1: 1.1 Signifiance_Sum: 2823 Power: 0.941"
## [1] "197 Sigma: 2 n: 20 beta_1: 1.2 Signifiance_Sum: 2912 Power: 0.970666666666667"
## [1] "198 Sigma: 2 n: 20 beta_1: 1.3 Signifiance_Sum: 2966 Power: 0.988666666666667"
## [1] "199 Sigma: 2 n: 20 beta_1: 1.4 Signifiance_Sum: 2990 Power: 0.996666666666667"
## [1] "200 Sigma: 2 n: 20 beta_1: 1.5 Signifiance_Sum: 2993 Power: 0.997666666666667"
## [1] "201 Sigma: 2 n: 20 beta_1: 1.6 Signifiance_Sum: 2999 Power: 0.999666666666667"
## [1] "202 Sigma: 2 n: 20 beta_1: 1.7 Signifiance_Sum: 3000 Power: 1"
## [1] "203 Sigma: 2 n: 20 beta_1: 1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "204 Sigma: 2 n: 20 beta_1: 1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "205 Sigma: 2 n: 20 beta_1: 2 Signifiance_Sum: 3000 Power: 1"
## [1] "206 Sigma: 2 n: 30 beta_1: -2 Signifiance_Sum: 3000 Power: 1"
## [1] "207 Sigma: 2 n: 30 beta_1: -1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "208 Sigma: 2 n: 30 beta_1: -1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "209 Sigma: 2 n: 30 beta_1: -1.7 Signifiance_Sum: 3000 Power: 1"
## [1] "210 Sigma: 2 n: 30 beta_1: -1.6 Signifiance_Sum: 3000 Power: 1"
## [1] "211 Sigma: 2 n: 30 beta_1: -1.5 Signifiance_Sum: 2999 Power: 0.999666666666667"
## [1] "212 Sigma: 2 n: 30 beta_1: -1.4 Signifiance_Sum: 2999 Power: 0.999666666666667"
## [1] "213 Sigma: 2 n: 30 beta_1: -1.3 Signifiance_Sum: 2995 Power: 0.998333333333333"
## [1] "214 Sigma: 2 n: 30 beta_1: -1.2 Signifiance_Sum: 2991 Power: 0.997"
## [1] "215 Sigma: 2 n: 30 beta_1: -1.1 Signifiance_Sum: 2964 Power: 0.988"
## [1] "216 Sigma: 2 n: 30 beta_1: -1 Signifiance_Sum: 2933 Power: 0.977666666666667"
## [1] "217 Sigma: 2 n: 30 beta_1: -0.9 Signifiance_Sum: 2811 Power: 0.937"
## [1] "218 Sigma: 2 n: 30 beta_1: -0.8 Signifiance_Sum: 2653 Power: 0.884333333333333"
## [1] "219 Sigma: 2 n: 30 beta_1: -0.7 Signifiance_Sum: 2386 Power: 0.795333333333333"
## [1] "220 Sigma: 2 n: 30 beta_1: -0.6 Signifiance_Sum: 1967 Power: 0.655666666666667"
## [1] "221 Sigma: 2 n: 30 beta_1: -0.5 Signifiance_Sum: 1466 Power: 0.488666666666667"
## [1] "222 Sigma: 2 n: 30 beta_1: -0.4 Signifiance_Sum: 1050 Power: 0.35"
## [1] "223 Sigma: 2 n: 30 beta_1: -0.3 Signifiance_Sum: 632 Power: 0.210666666666667"
## [1] "224 Sigma: 2 n: 30 beta_1: -0.2 Signifiance_Sum: 385 Power: 0.128333333333333"
## [1] "225 Sigma: 2 n: 30 beta_1: -0.0999999999999999 Signifiance_Sum: 194 Power: 0.0646666666666667"
## [1] "226 Sigma: 2 n: 30 beta_1: 0 Signifiance_Sum: 129 Power: 0.043"
## [1] "227 Sigma: 2 n: 30 beta_1: 0.1 Signifiance_Sum: 181 Power: 0.0603333333333333"
## [1] "228 Sigma: 2 n: 30 beta_1: 0.2 Signifiance_Sum: 388 Power: 0.129333333333333"
## [1] "229 Sigma: 2 n: 30 beta_1: 0.3 Signifiance_Sum: 650 Power: 0.216666666666667"
## [1] "230 Sigma: 2 n: 30 beta_1: 0.4 Signifiance_Sum: 1014 Power: 0.338"
## [1] "231 Sigma: 2 n: 30 beta_1: 0.5 Signifiance_Sum: 1490 Power: 0.496666666666667"
## [1] "232 Sigma: 2 n: 30 beta_1: 0.6 Signifiance_Sum: 1963 Power: 0.654333333333333"
## [1] "233 Sigma: 2 n: 30 beta_1: 0.7 Signifiance_Sum: 2371 Power: 0.790333333333333"
## [1] "234 Sigma: 2 n: 30 beta_1: 0.8 Signifiance_Sum: 2637 Power: 0.879"
## [1] "235 Sigma: 2 n: 30 beta_1: 0.9 Signifiance_Sum: 2857 Power: 0.952333333333333"
## [1] "236 Sigma: 2 n: 30 beta_1: 1 Signifiance_Sum: 2943 Power: 0.981"
## [1] "237 Sigma: 2 n: 30 beta_1: 1.1 Signifiance_Sum: 2971 Power: 0.990333333333333"
## [1] "238 Sigma: 2 n: 30 beta_1: 1.2 Signifiance_Sum: 2995 Power: 0.998333333333333"
## [1] "239 Sigma: 2 n: 30 beta_1: 1.3 Signifiance_Sum: 2998 Power: 0.999333333333333"
## [1] "240 Sigma: 2 n: 30 beta_1: 1.4 Signifiance_Sum: 2998 Power: 0.999333333333333"
## [1] "241 Sigma: 2 n: 30 beta_1: 1.5 Signifiance_Sum: 3000 Power: 1"
## [1] "242 Sigma: 2 n: 30 beta_1: 1.6 Signifiance_Sum: 3000 Power: 1"
## [1] "243 Sigma: 2 n: 30 beta_1: 1.7 Signifiance_Sum: 3000 Power: 1"
## [1] "244 Sigma: 2 n: 30 beta_1: 1.8 Signifiance_Sum: 3000 Power: 1"
## [1] "245 Sigma: 2 n: 30 beta_1: 1.9 Signifiance_Sum: 3000 Power: 1"
## [1] "246 Sigma: 2 n: 30 beta_1: 2 Signifiance_Sum: 3000 Power: 1"
## [1] "247 Sigma: 4 n: 10 beta_1: -2 Signifiance_Sum: 1790 Power: 0.596666666666667"
## [1] "248 Sigma: 4 n: 10 beta_1: -1.9 Signifiance_Sum: 1657 Power: 0.552333333333333"
## [1] "249 Sigma: 4 n: 10 beta_1: -1.8 Signifiance_Sum: 1543 Power: 0.514333333333333"
## [1] "250 Sigma: 4 n: 10 beta_1: -1.7 Signifiance_Sum: 1428 Power: 0.476"
## [1] "251 Sigma: 4 n: 10 beta_1: -1.6 Signifiance_Sum: 1273 Power: 0.424333333333333"
## [1] "252 Sigma: 4 n: 10 beta_1: -1.5 Signifiance_Sum: 1182 Power: 0.394"
## [1] "253 Sigma: 4 n: 10 beta_1: -1.4 Signifiance_Sum: 1026 Power: 0.342"
## [1] "254 Sigma: 4 n: 10 beta_1: -1.3 Signifiance_Sum: 919 Power: 0.306333333333333"
## [1] "255 Sigma: 4 n: 10 beta_1: -1.2 Signifiance_Sum: 798 Power: 0.266"
## [1] "256 Sigma: 4 n: 10 beta_1: -1.1 Signifiance_Sum: 692 Power: 0.230666666666667"
## [1] "257 Sigma: 4 n: 10 beta_1: -1 Signifiance_Sum: 580 Power: 0.193333333333333"
## [1] "258 Sigma: 4 n: 10 beta_1: -0.9 Signifiance_Sum: 517 Power: 0.172333333333333"
## [1] "259 Sigma: 4 n: 10 beta_1: -0.8 Signifiance_Sum: 443 Power: 0.147666666666667"
## [1] "260 Sigma: 4 n: 10 beta_1: -0.7 Signifiance_Sum: 364 Power: 0.121333333333333"
## [1] "261 Sigma: 4 n: 10 beta_1: -0.6 Signifiance_Sum: 323 Power: 0.107666666666667"
## [1] "262 Sigma: 4 n: 10 beta_1: -0.5 Signifiance_Sum: 285 Power: 0.095"
## [1] "263 Sigma: 4 n: 10 beta_1: -0.4 Signifiance_Sum: 247 Power: 0.0823333333333333"
## [1] "264 Sigma: 4 n: 10 beta_1: -0.3 Signifiance_Sum: 196 Power: 0.0653333333333333"
## [1] "265 Sigma: 4 n: 10 beta_1: -0.2 Signifiance_Sum: 155 Power: 0.0516666666666667"
## [1] "266 Sigma: 4 n: 10 beta_1: -0.0999999999999999 Signifiance_Sum: 162 Power: 0.054"
## [1] "267 Sigma: 4 n: 10 beta_1: 0 Signifiance_Sum: 149 Power: 0.0496666666666667"
## [1] "268 Sigma: 4 n: 10 beta_1: 0.1 Signifiance_Sum: 133 Power: 0.0443333333333333"
## [1] "269 Sigma: 4 n: 10 beta_1: 0.2 Signifiance_Sum: 176 Power: 0.0586666666666667"
## [1] "270 Sigma: 4 n: 10 beta_1: 0.3 Signifiance_Sum: 184 Power: 0.0613333333333333"
## [1] "271 Sigma: 4 n: 10 beta_1: 0.4 Signifiance_Sum: 233 Power: 0.0776666666666667"
## [1] "272 Sigma: 4 n: 10 beta_1: 0.5 Signifiance_Sum: 273 Power: 0.091"
## [1] "273 Sigma: 4 n: 10 beta_1: 0.6 Signifiance_Sum: 302 Power: 0.100666666666667"
## [1] "274 Sigma: 4 n: 10 beta_1: 0.7 Signifiance_Sum: 334 Power: 0.111333333333333"
## [1] "275 Sigma: 4 n: 10 beta_1: 0.8 Signifiance_Sum: 431 Power: 0.143666666666667"
## [1] "276 Sigma: 4 n: 10 beta_1: 0.9 Signifiance_Sum: 501 Power: 0.167"
## [1] "277 Sigma: 4 n: 10 beta_1: 1 Signifiance_Sum: 597 Power: 0.199"
## [1] "278 Sigma: 4 n: 10 beta_1: 1.1 Signifiance_Sum: 707 Power: 0.235666666666667"
## [1] "279 Sigma: 4 n: 10 beta_1: 1.2 Signifiance_Sum: 828 Power: 0.276"
## [1] "280 Sigma: 4 n: 10 beta_1: 1.3 Signifiance_Sum: 931 Power: 0.310333333333333"
## [1] "281 Sigma: 4 n: 10 beta_1: 1.4 Signifiance_Sum: 1028 Power: 0.342666666666667"
## [1] "282 Sigma: 4 n: 10 beta_1: 1.5 Signifiance_Sum: 1131 Power: 0.377"
## [1] "283 Sigma: 4 n: 10 beta_1: 1.6 Signifiance_Sum: 1264 Power: 0.421333333333333"
## [1] "284 Sigma: 4 n: 10 beta_1: 1.7 Signifiance_Sum: 1371 Power: 0.457"
## [1] "285 Sigma: 4 n: 10 beta_1: 1.8 Signifiance_Sum: 1531 Power: 0.510333333333333"
## [1] "286 Sigma: 4 n: 10 beta_1: 1.9 Signifiance_Sum: 1643 Power: 0.547666666666667"
## [1] "287 Sigma: 4 n: 10 beta_1: 2 Signifiance_Sum: 1820 Power: 0.606666666666667"
## [1] "288 Sigma: 4 n: 20 beta_1: -2 Signifiance_Sum: 2686 Power: 0.895333333333333"
## [1] "289 Sigma: 4 n: 20 beta_1: -1.9 Signifiance_Sum: 2611 Power: 0.870333333333333"
## [1] "290 Sigma: 4 n: 20 beta_1: -1.8 Signifiance_Sum: 2474 Power: 0.824666666666667"
## [1] "291 Sigma: 4 n: 20 beta_1: -1.7 Signifiance_Sum: 2358 Power: 0.786"
## [1] "292 Sigma: 4 n: 20 beta_1: -1.6 Signifiance_Sum: 2180 Power: 0.726666666666667"
## [1] "293 Sigma: 4 n: 20 beta_1: -1.5 Signifiance_Sum: 2012 Power: 0.670666666666667"
## [1] "294 Sigma: 4 n: 20 beta_1: -1.4 Signifiance_Sum: 1803 Power: 0.601"
## [1] "295 Sigma: 4 n: 20 beta_1: -1.3 Signifiance_Sum: 1662 Power: 0.554"
## [1] "296 Sigma: 4 n: 20 beta_1: -1.2 Signifiance_Sum: 1469 Power: 0.489666666666667"
## [1] "297 Sigma: 4 n: 20 beta_1: -1.1 Signifiance_Sum: 1311 Power: 0.437"
## [1] "298 Sigma: 4 n: 20 beta_1: -1 Signifiance_Sum: 1074 Power: 0.358"
## [1] "299 Sigma: 4 n: 20 beta_1: -0.9 Signifiance_Sum: 926 Power: 0.308666666666667"
## [1] "300 Sigma: 4 n: 20 beta_1: -0.8 Signifiance_Sum: 763 Power: 0.254333333333333"
## [1] "301 Sigma: 4 n: 20 beta_1: -0.7 Signifiance_Sum: 601 Power: 0.200333333333333"
## [1] "302 Sigma: 4 n: 20 beta_1: -0.6 Signifiance_Sum: 533 Power: 0.177666666666667"
## [1] "303 Sigma: 4 n: 20 beta_1: -0.5 Signifiance_Sum: 383 Power: 0.127666666666667"
## [1] "304 Sigma: 4 n: 20 beta_1: -0.4 Signifiance_Sum: 303 Power: 0.101"
## [1] "305 Sigma: 4 n: 20 beta_1: -0.3 Signifiance_Sum: 251 Power: 0.0836666666666667"
## [1] "306 Sigma: 4 n: 20 beta_1: -0.2 Signifiance_Sum: 164 Power: 0.0546666666666667"
## [1] "307 Sigma: 4 n: 20 beta_1: -0.0999999999999999 Signifiance_Sum: 194 Power: 0.0646666666666667"
## [1] "308 Sigma: 4 n: 20 beta_1: 0 Signifiance_Sum: 133 Power: 0.0443333333333333"
## [1] "309 Sigma: 4 n: 20 beta_1: 0.1 Signifiance_Sum: 159 Power: 0.053"
## [1] "310 Sigma: 4 n: 20 beta_1: 0.2 Signifiance_Sum: 198 Power: 0.066"
## [1] "311 Sigma: 4 n: 20 beta_1: 0.3 Signifiance_Sum: 219 Power: 0.073"
## [1] "312 Sigma: 4 n: 20 beta_1: 0.4 Signifiance_Sum: 309 Power: 0.103"
## [1] "313 Sigma: 4 n: 20 beta_1: 0.5 Signifiance_Sum: 360 Power: 0.12"
## [1] "314 Sigma: 4 n: 20 beta_1: 0.6 Signifiance_Sum: 482 Power: 0.160666666666667"
## [1] "315 Sigma: 4 n: 20 beta_1: 0.7 Signifiance_Sum: 591 Power: 0.197"
## [1] "316 Sigma: 4 n: 20 beta_1: 0.8 Signifiance_Sum: 729 Power: 0.243"
## [1] "317 Sigma: 4 n: 20 beta_1: 0.9 Signifiance_Sum: 916 Power: 0.305333333333333"
## [1] "318 Sigma: 4 n: 20 beta_1: 1 Signifiance_Sum: 1120 Power: 0.373333333333333"
## [1] "319 Sigma: 4 n: 20 beta_1: 1.1 Signifiance_Sum: 1280 Power: 0.426666666666667"
## [1] "320 Sigma: 4 n: 20 beta_1: 1.2 Signifiance_Sum: 1451 Power: 0.483666666666667"
## [1] "321 Sigma: 4 n: 20 beta_1: 1.3 Signifiance_Sum: 1660 Power: 0.553333333333333"
## [1] "322 Sigma: 4 n: 20 beta_1: 1.4 Signifiance_Sum: 1841 Power: 0.613666666666667"
## [1] "323 Sigma: 4 n: 20 beta_1: 1.5 Signifiance_Sum: 1993 Power: 0.664333333333333"
## [1] "324 Sigma: 4 n: 20 beta_1: 1.6 Signifiance_Sum: 2171 Power: 0.723666666666667"
## [1] "325 Sigma: 4 n: 20 beta_1: 1.7 Signifiance_Sum: 2349 Power: 0.783"
## [1] "326 Sigma: 4 n: 20 beta_1: 1.8 Signifiance_Sum: 2406 Power: 0.802"
## [1] "327 Sigma: 4 n: 20 beta_1: 1.9 Signifiance_Sum: 2597 Power: 0.865666666666667"
## [1] "328 Sigma: 4 n: 20 beta_1: 2 Signifiance_Sum: 2654 Power: 0.884666666666667"
## [1] "329 Sigma: 4 n: 30 beta_1: -2 Signifiance_Sum: 2923 Power: 0.974333333333333"
## [1] "330 Sigma: 4 n: 30 beta_1: -1.9 Signifiance_Sum: 2898 Power: 0.966"
## [1] "331 Sigma: 4 n: 30 beta_1: -1.8 Signifiance_Sum: 2835 Power: 0.945"
## [1] "332 Sigma: 4 n: 30 beta_1: -1.7 Signifiance_Sum: 2759 Power: 0.919666666666667"
## [1] "333 Sigma: 4 n: 30 beta_1: -1.6 Signifiance_Sum: 2632 Power: 0.877333333333333"
## [1] "334 Sigma: 4 n: 30 beta_1: -1.5 Signifiance_Sum: 2496 Power: 0.832"
## [1] "335 Sigma: 4 n: 30 beta_1: -1.4 Signifiance_Sum: 2378 Power: 0.792666666666667"
## [1] "336 Sigma: 4 n: 30 beta_1: -1.3 Signifiance_Sum: 2190 Power: 0.73"
## [1] "337 Sigma: 4 n: 30 beta_1: -1.2 Signifiance_Sum: 1966 Power: 0.655333333333333"
## [1] "338 Sigma: 4 n: 30 beta_1: -1.1 Signifiance_Sum: 1753 Power: 0.584333333333333"
## [1] "339 Sigma: 4 n: 30 beta_1: -1 Signifiance_Sum: 1534 Power: 0.511333333333333"
## [1] "340 Sigma: 4 n: 30 beta_1: -0.9 Signifiance_Sum: 1292 Power: 0.430666666666667"
## [1] "341 Sigma: 4 n: 30 beta_1: -0.8 Signifiance_Sum: 1059 Power: 0.353"
## [1] "342 Sigma: 4 n: 30 beta_1: -0.7 Signifiance_Sum: 855 Power: 0.285"
## [1] "343 Sigma: 4 n: 30 beta_1: -0.6 Signifiance_Sum: 672 Power: 0.224"
## [1] "344 Sigma: 4 n: 30 beta_1: -0.5 Signifiance_Sum: 477 Power: 0.159"
## [1] "345 Sigma: 4 n: 30 beta_1: -0.4 Signifiance_Sum: 363 Power: 0.121"
## [1] "346 Sigma: 4 n: 30 beta_1: -0.3 Signifiance_Sum: 264 Power: 0.088"
## [1] "347 Sigma: 4 n: 30 beta_1: -0.2 Signifiance_Sum: 192 Power: 0.064"
## [1] "348 Sigma: 4 n: 30 beta_1: -0.0999999999999999 Signifiance_Sum: 163 Power: 0.0543333333333333"
## [1] "349 Sigma: 4 n: 30 beta_1: 0 Signifiance_Sum: 147 Power: 0.049"
## [1] "350 Sigma: 4 n: 30 beta_1: 0.1 Signifiance_Sum: 190 Power: 0.0633333333333333"
## [1] "351 Sigma: 4 n: 30 beta_1: 0.2 Signifiance_Sum: 227 Power: 0.0756666666666667"
## [1] "352 Sigma: 4 n: 30 beta_1: 0.3 Signifiance_Sum: 243 Power: 0.081"
## [1] "353 Sigma: 4 n: 30 beta_1: 0.4 Signifiance_Sum: 389 Power: 0.129666666666667"
## [1] "354 Sigma: 4 n: 30 beta_1: 0.5 Signifiance_Sum: 514 Power: 0.171333333333333"
## [1] "355 Sigma: 4 n: 30 beta_1: 0.6 Signifiance_Sum: 655 Power: 0.218333333333333"
## [1] "356 Sigma: 4 n: 30 beta_1: 0.7 Signifiance_Sum: 883 Power: 0.294333333333333"
## [1] "357 Sigma: 4 n: 30 beta_1: 0.8 Signifiance_Sum: 1055 Power: 0.351666666666667"
## [1] "358 Sigma: 4 n: 30 beta_1: 0.9 Signifiance_Sum: 1266 Power: 0.422"
## [1] "359 Sigma: 4 n: 30 beta_1: 1 Signifiance_Sum: 1544 Power: 0.514666666666667"
## [1] "360 Sigma: 4 n: 30 beta_1: 1.1 Signifiance_Sum: 1741 Power: 0.580333333333333"
## [1] "361 Sigma: 4 n: 30 beta_1: 1.2 Signifiance_Sum: 2037 Power: 0.679"
## [1] "362 Sigma: 4 n: 30 beta_1: 1.3 Signifiance_Sum: 2200 Power: 0.733333333333333"
## [1] "363 Sigma: 4 n: 30 beta_1: 1.4 Signifiance_Sum: 2337 Power: 0.779"
## [1] "364 Sigma: 4 n: 30 beta_1: 1.5 Signifiance_Sum: 2521 Power: 0.840333333333333"
## [1] "365 Sigma: 4 n: 30 beta_1: 1.6 Signifiance_Sum: 2665 Power: 0.888333333333333"
## [1] "366 Sigma: 4 n: 30 beta_1: 1.7 Signifiance_Sum: 2755 Power: 0.918333333333333"
## [1] "367 Sigma: 4 n: 30 beta_1: 1.8 Signifiance_Sum: 2824 Power: 0.941333333333333"
## [1] "368 Sigma: 4 n: 30 beta_1: 1.9 Signifiance_Sum: 2885 Power: 0.961666666666667"
## [1] "369 Sigma: 4 n: 30 beta_1: 2 Signifiance_Sum: 2946 Power: 0.982"
par(mfrow=c(1, 2))
plot(model_sim[,1],model_sim[,2], type="o", col="dodgerblue", pch=1, lty=1,main="1000 Iterations",xlab = "beta_1",ylab = "Power")
points(model_sim[,1], model_sim[,3], col="red", pch=2,lty=2)
lines(model_sim[,1], model_sim[,3], col="red",lty=2)
points(model_sim[,1], model_sim[,4], col="tan1", pch=3,lty=3)
lines(model_sim[,1], model_sim[,4], col="tan1",lty=3)
points(model_sim[,1],model_sim[,5], type="o", col="yellow", pch=4, lty=4)
lines(model_sim[,1],model_sim[,5], type="o", col="yellow", pch=4, lty=4)
points(model_sim[,1], model_sim[,6], col="deeppink", pch=5,lty=5)
lines(model_sim[,1], model_sim[,6], col="deeppink",lty=5)
points(model_sim[,1], model_sim[,7], col="darkolivegreen4", pch=6,lty=6)
lines(model_sim[,1], model_sim[,7], col="darkolivegreen4",lty=6)
points(model_sim[,1],model_sim[,8], type="o", col="darkgreen", pch=7, lty=7)
lines(model_sim[,1],model_sim[,8], type="o", col="darkgreen", pch=7, lty=7)
points(model_sim[,1], model_sim[,9], col="darkblue", pch=8,lty=8)
lines(model_sim[,1], model_sim[,9], col="darkblue",lty=8)
points(model_sim[,1], model_sim[,10], col="blueviolet", pch=9,lty=9)
lines(model_sim[,1], model_sim[,10], col="blueviolet",lty=9)
legend("topright",legend=c("σ=1,n=10","σ=1,n=20","σ=1,n=30","σ=2,n=10","σ=2,n=20","σ=2,n=30","σ=3,n=10","σ=3,n=20","σ=3,n=30"),col=c("dodgerblue","red","darkorchid","yellow","deeppink","darkolivegreen4","darkgreen","darkblue","blueviolet"),pch=c(1,2,3,4,5,6,7,8,9),lty=c(1,2,3,4,5,6,7,8,9))
#Below code is to plot the simulated beta_1 for 3000 simulations. This plot will helps us to understands, whether there is anything changes because of increase the simulation
#from 1000 iteration to 3000 iteration
plot(model_sim_3000[,1],model_sim_3000[,2], type="o", col="dodgerblue", pch=1, lty=1,main="3000 Iterations",xlab = "beta_1",ylab = "Power")
points(model_sim_3000[,1], model_sim_3000[,3], col="red", pch=2,lty=2)
lines(model_sim_3000[,1], model_sim_3000[,3], col="red",lty=2)
points(model_sim_3000[,1], model_sim_3000[,4], col="tan1", pch=3,lty=3)
lines(model_sim_3000[,1], model_sim_3000[,4], col="tan1",lty=3)
points(model_sim_3000[,1],model_sim_3000[,5], type="o", col="yellow", pch=4, lty=4)
lines(model_sim_3000[,1],model_sim_3000[,5], type="o", col="yellow", pch=4, lty=4)
points(model_sim_3000[,1], model_sim_3000[,6], col="deeppink", pch=5,lty=5)
lines(model_sim_3000[,1], model_sim_3000[,6], col="deeppink",lty=5)
points(model_sim_3000[,1], model_sim_3000[,7], col="darkolivegreen4", pch=6,lty=6)
lines(model_sim_3000[,1], model_sim_3000[,7], col="darkolivegreen4",lty=6)
points(model_sim_3000[,1],model_sim_3000[,8], type="o", col="darkgreen", pch=7, lty=7)
lines(model_sim_3000[,1],model_sim_3000[,8], type="o", col="darkgreen", pch=7, lty=7)
points(model_sim_3000[,1], model_sim_3000[,9], col="darkblue", pch=8,lty=8)
lines(model_sim_3000[,1], model_sim_3000[,9], col="darkblue",lty=8)
points(model_sim_3000[,1], model_sim_3000[,10], col="blueviolet", pch=9,lty=9)
lines(model_sim_3000[,1], model_sim_3000[,10], col="blueviolet",lty=9)
legend("topright",legend=c("σ=1,n=10","σ=1,n=20","σ=1,n=30","σ=2,n=10","σ=2,n=20","σ=2,n=30","σ=3,n=10","σ=3,n=20","σ=3,n=30"),col=c("dodgerblue","red","darkorchid","yellow","deeppink","darkolivegreen4","darkgreen","darkblue","blueviolet"),pch=c(1,2,3,4,5,6,7,8,9),lty=c(1,2,3,4,5,6,7,8,9))
From the above plots its clearly visible that the plots for the 3000 iterations have much smoother curve with respect to the Power value, as compared to the 1000 iterations (where the power value is ups-down). Let’s take the range of the Power for the 1000 vs 3000 iterations for a given value of sigma (let’s say , sigma = 1) and for a particular value of sample size ( n= 10).
From the range prospective the 3000 iterative have slightly increased value as compared to 1000 iteration. Hence if we increase the iteration, the Power becomes more and more accurate to the Truth, hence I conclude that 1000 iteration is sufficient to know the impact of beta(β1), sample size(n) and sigma(σ) on the Power, however 1000 iteration is not sufficient to know the truth of Power